Answer:
It will take about 35.49 hours for the water to leak out of the barrel.
Step-by-step explanation:
Let [tex]y(t)[/tex] be the depth of water in the barrel at time [tex]t[/tex], where [tex]y[/tex] is measured in inches and [tex]t[/tex] in hours.
We know that water is leaking out of a large barrel at a rate proportional to the square root of the depth of the water at that time. We then have that
[tex]\frac{dy}{dt}=-k\sqrt{y}[/tex]
where [tex]k[/tex] is a constant of proportionality.
Separation of variables is a common method for solving differential equations. To solve the above differential equation you must:
Multiply by [tex]\frac{1}{\sqrt{y}}[/tex]
[tex]\frac{1}{\sqrt{y}}\frac{dy}{dt}=-k[/tex]
Multiply by [tex]dt[/tex]
[tex]\frac{1}{\sqrt{y}}\cdot dy=-k\cdot dt[/tex]
Take integral
[tex]\int \frac{1}{\sqrt{y}}\cdot dy=\int-k\cdot dt[/tex]
Integrate
[tex]2\sqrt{y}=-kt+C[/tex]
Isolate [tex]y[/tex]
[tex]y(t)=(\frac{C}{2} -\frac{k}{2}t)^2[/tex]
We know that the water level starts at 36, this means [tex]y(0)=36[/tex]. We use this information to find the value of [tex]C[/tex].
[tex]36=(\frac{C}{2} -\frac{k}{2}(0))^2\\C=12[/tex]
[tex]y(t)=(\frac{12}{2} -\frac{k}{2}t)^2\\\\y(t)=(6 -\frac{k}{2}t)^2[/tex]
At t = 1, y = 34
[tex]34=(6 -\frac{k}{2}(1))^2\\k=12-2\sqrt{34}[/tex]
So our formula for the depth of water in the barrel is
[tex]y(t)=(6 -\frac{12-2\sqrt{34}}{2}t)^2\\\\y(t)=\left(6-\left(6-\sqrt{34}\right)t\right)^2\\[/tex]
To find the time, [tex]t[/tex], at which all the water leaks out of the barrel, we solve the equation
[tex]\left(6-\left(6-\sqrt{34}\right)t\right)^2=0\\\\t=3\left(6+\sqrt{34}\right)\approx 35.49[/tex]
Thus, it will take about 35.49 hours for the water to leak out of the barrel.
The time it will take for all of the water to drain out of that barrel is 35.5 hours approx.
What is directly proportional and inversely proportional relationship?Let there are two variables p and q
Then, p and q are said to be directly proportional to each other if
[tex]p = kq[/tex]
where k is some constant number called constant of proportionality.
This directly proportional relationship between p and q is written as
[tex]p \propto q[/tex] where that middle sign is the sign of proportionality.
In a directly proportional relationship, increasing one variable will increase another.
Now let m and n are two variables.
Then m and n are said to be inversely proportional to each other if
[tex]m = \dfrac{c}{n} \\\\ \text{or} \\\\ n = \dfrac{c}{m}[/tex]
(both are equal)
where c is a constant number called constant of proportionality.
This inversely proportional relationship is denoted by
[tex]m \propto \dfrac{1}{n} \\\\ \text{or} \\\\n \propto \dfrac{1}{m}[/tex]
As visible, increasing one variable will decrease the other variable if both are inversely proportional.
For the considered case, let we take three variables as:
t = time passed since start (in hours)[tex]x[/tex] = depth of water at time 't'[tex]y[/tex] = amount of water(in terms of depth) leaking per hour, in litersNow, when t = 0, x = 36 inches.
and at t = 1, x = 34 inches.
So depth of water is function of time passed. Let x = f(t)
Also, y is negative rate of change of x with respect to t(since depth is decreasing, and draining is measuring decrement rate, thus, negative of increment rate), or
[tex]y = -\dfrac{dx}{dt}[/tex]
We're given that: "Water is leaking out of a large barrel at a rate proportional to the square root of the depth of the water at that time"
That means, [tex]y \propto \sqrt{x}[/tex]
Let the constant of proportionality be k, then,
[tex]y = \sqrt{x}[/tex]
Since we've [tex]y = -\dfrac{dx}{dt}[/tex], therefore,
[tex]-\dfrac{dx}{dt} = k\sqrt{x}\\\\-\dfrac{dx}{\sqrt{x}} = kdt\\\\\text{Integrating both the sides, we get}\\\\\int -\dfrac{dx}{\sqrt{x}} = \int kdt\\\\-2\sqrt{x} = kt + C[/tex]
where C is integration constant.
Since at t = 0 hours passed, x = 36 inches, and at t = 1 hour passed, x = 34 inches, we get two equations as:
[tex]-2\sqrt{x} = kt + C\\-2\sqrt{36} = -12 = C\\-2\sqrt{34} = k + C[/tex]
Putting value of C from first equation in second, we get:
[tex]k = -2\sqrt{34} + C \\k = -2\sqrt{34} - 12[/tex]
Therefore, the relationship between depth of water and time we get is:
[tex]-2\sqrt{x} = (-2\sqrt{34} - 12)t -12\\\sqrt{x} = (\sqrt{34} - 6)t + 6[/tex]
When the whole barrel gets empty, the depth of water becomes 0. The time for it is calculated using above equation as:
[tex]\sqrt{x} = (\sqrt{34} - 6)t + 6\\0 = (\sqrt{34} - 6)t + 6\\t = \dfrac{6}{6 - \sqrt{34}} \approx 35.5[/tex](in hours)
Thus, the time it will take for all of the water to drain out of that barrel is 35.5 hours approx.
Learn more about differential equations here:
https://brainly.com/question/14658115
Geoff wants to purchase a circular area rug for his room, but he can't decide between the small size and the large size. A small rug covers an area of approximately 67 square feet. If a large size rug has twice the dimensions of a small rug, how many times larger is the large rug?
Answer: The larger rug is 4 times larger
Step-by-step explanation:
The formula for determining the area of a circle is expressed as
Area = πr²
Where
r represents the radius of the circle.
π is a constant whose value is 3.14
The small circular rug covers an area of approximately 67 square feet. This means that
67 = πr²
r² = 67/π = 67/3.14 = 21.34
r = √21.34 = 4.62
If a large size rug has twice the dimensions of a small rug, it means that the radius of the larger rug would be
2 × 4.62 = 9.24
The area of the larger rug would be
3.14 × 9.24² = 268.06ft²
Therefore
268.06/67 = 4
The larger rug is 4 times larger
Given triangle ABC with altitude labeled x. Angles ADB and CDB are right angles by _____1._____, making triangle ABD and triangle BCD right triangles. Using the trigonometric ratios and . Multiplying to isolate x in both equations gives x = _____2._____ and x = a ⋅ sinC. We also know that x = x by the reflexive property. By the substitution property, _____3._____. Dividing each side of the equation by ac gives: .
Answer:
(1) ADB = CDB = 90°
(2) c sinA
(3) (sinA)/a = (sinB)/b
Step-by-step explanation:
1. ADB = CDB = 90°
2. c sinA
since,
sin A = x/c , sin C = x/a
so x = c sinA and a sinC
3. from reflective property of x,
since x = c sinA
and x = a sinC
we substitute each equivalently
that is,
c sinA = a sinC
dividing each sides of the equation by ac we have ,
(c sinA)/ac = ( a sinC)/ac
simplifying we have,
(sinA)/a = (sinB)/b
Therefore the above equation is referred to as the SINE RULE.
Aidan is running for student body president. In today's election, he received 462 votes out of a total of 825 votes cast. What percent of votes did Aidan get? %
Answer:
Aidan got 56% of votes.
Step-by-step explanation:
Given:
Aidan is running for student body president. In today's election, he received 462 votes out of a total of 825 votes cast.
Now, to find the percent of votes did Aidan get.
Total votes cast = 825 votes.
Votes he received = 462 votes.
Now, to get the percent of votes Aidan get:
[tex]\frac{Votes\ he\ received}{Total\ votes\ cast} \times 100[/tex]
[tex]=\frac{462}{825} \times 100[/tex]
[tex]=0.56\times 100[/tex]
[tex]=56\%.[/tex]
Therefore, Aidan got 56% of votes.
Answer:
56%
Step-by-step explanation:
Aidan got fifty-six percent of the votes.
To solve this problem, start by letting p represent the unknown percent and write the percent as the fraction p over one hundred.
Write a second ratio that expresses the part-whole relationship between the numbers four hundred sixty-two over eight hundred twenty-five.
Set up a proportion between the two ratios.
Write the cross product equation. Eight hundred twenty-five p equals one hundred times four hundred sixty-two, or eight hundred twenty-five p equals forty-six thousand two hundred.
Divide each side of the equation by eight hundred twenty-five to solve for p.
p equals fifty-six.
Aidan received fifty-six percent of the votes cast.
Leon wants to estimate the height of a building. Leon's eyes are 6 feet above ground. He stands 25 feet from the building and sights the top of the building at a 77° angle of elevation. What is the building's height to the nearest tenth of a foot?
Answer:
114.29 ft
Step-by-step explanation:
tan ∅ = Opp/Adj
tan 77 = x/25
x = 25 tan 77
x = 108.29ft
Plus 6ft = 114.29 ft
When large bags of candies are packaged, the number of candies in each bag must be within 4 of 120 pieces. Write an absolute value equation to represent p
Answer:
Step-by-step explanation:
p = 120 ± 4
An escalator at a shopping center is 200ft and 9in long, and rises at an angle of 15 degrees. What is the vertical rise of the escalator. Round to the nearest inch
Answer: 623 inches
Step-by-step explanation:
We can model this escalator as a right triangle, in which its length is the hypotenuse and its vertical rise ([tex]x[/tex]) is on of the sides of the triangle (as shown in the figure).
So, if we want to find [tex]x[/tex] we have to use the trigonometric function sine:
[tex]sin(15\°)=\frac{opposite-side}{hypotenuse}[/tex] (1)
Here, the opposite side is [tex]x[/tex] and the hypotenuse is [tex]200 ft 9 in[/tex].
Now we have to transform [tex]200 ft 9 in[/tex] to inches, in this case we only have to convert [tex]200 ft[/tex] to inches, knowing that [tex]1 ft=12 in[/tex]:
[tex]200 ft \frac{12 in}{1 ft}=2400 in[/tex]
[tex]200 ft + 9 in=2400 in+ 9 in=2409 in[/tex]
Substituting this value in (1):
[tex]sin(15\°)=\frac{x}{2409 in}[/tex] (2)
Isolating [tex]x[/tex]:
[tex]x=frac{sin(15\°)}{2409 in}[/tex]
Finally:
[tex]x=623.49 in \approx 623 in[/tex]
Jaylan and casper are equal partners in J&C racoon hats. Jaylan contributed 12000 and casper contributed inventory with a FMV of 12000 and an adjusted basis of 10000. What is each partnership basis
Answer:
The question continues ; Which of the following is true ;
Jaylan basis in the partnership is $12,00 and Caspers is 10,000
Jaylan and Casper each have a basis in the partnership of 12,000
Jaylan will have a larger share of the profits than Casper
The first and third answers are both correct
Answer ; Jaylan and Casper each have a basis in the partnership of 12,000
Step-by-step explanation:
Both Jaylan and casper will have a partnership basis of $12,000 , it was mentioned initially in the question that (Jaylan and casper are equal partners in J&C racoon hats), equal partners implies both Jaylan and casper with have equal profit and equal losses as the case maybe Irrespective of the amount each contributed. Hence Jaylan and Casper each have a basis in the partnership of 12,000.
The present above is a 10 in by 10 in by 10 in cube. How many square inches of wrapping paper do you need to wrap the box?
Answer:
600 in^2.
Step-by-step explanation:
There are 6 faces on a cube so the area we need is:
6 * 10 * 10
= 600 in^2.
suppose sin(a)=3/4
use the trig identity sin^2(a)+cos^2(a)=1
and the trig identity tan(a)=sin(a)/cos(a)
to find tan (a) in quad II.
Round to the nearest hundredth.
well, we know the sine, and we also know that we're on the II Quadrant, let's recall that on the II Quadrant sine is positive whilst cosine is negative.
[tex]\bf sin^2(\theta)+cos^2(\theta)=1~\hspace{10em} tan(\theta )=\cfrac{sin(\theta )}{cos(\theta )} \\\\[-0.35em] ~\dotfill\\\\ sin^2(a)+cos^2(a)=1\implies cos^2(a) = 1-sin^2(a) \\\\\\ cos^2(a) = 1-[sin(a)]^2\implies cos^2(a) = 1-\left( \cfrac{3}{4} \right)^2\implies cos^2(a) = 1-\cfrac{3^2}{4^2} \\\\\\ cos^2(a) = 1-\cfrac{9}{16}\implies cos^2(a) = \cfrac{7}{16}\implies cos(a)=\pm\sqrt{\cfrac{7}{16}}[/tex]
[tex]\bf cos(a)=\pm\cfrac{\sqrt{7}}{\sqrt{16}}\implies cos(a)=\pm\cfrac{\sqrt{7}}{4}\implies \stackrel{\textit{on the II Quadrant}}{cos(a)=-\cfrac{\sqrt{7}}{4}}\\\\[-0.35em]~\dotfill\\\\tan(a)=\cfrac{sin(a)}{cos(a)}\implies tan(a)=\cfrac{~~\frac{3}{4}~~}{-\frac{\sqrt{7}}{4}}\implies tan(a)=\cfrac{3}{4}\cdot \cfrac{4}{-\sqrt{7}}\\\\\\tan(a)=-\cfrac{3}{\sqrt{7}}\implies \stackrel{\textit{rounded up}}{tan(a) = -1.13}[/tex]
Rebeca spent $32.55 for a photo album and three identical candles. The photo album cost $17.50 and the sales tax was $1.55. How much did each candle cost
Answer: $4.50 per candle
Step-by-step explanation:
$32.55 - $17.50 - $1.55 = $13.50 for all the candles
To find the price of a single candle we divide our answer by 3
13.50/3 = $4.50
25.) If y varies directly as x, and y = -18 as x = -2, find y for the x-value of 20.
Answer: the value of y is 180
Step-by-step explanation:
If y varies directly as x, then as y increases,x increases and as y decreases, x decreases.
We would introduce a constant of proportionality, k. Therefore,
y = kx
When y = - 18, x = - 2
Therefore,
- 18 = - 2 × k
Dividing the left hand side and the right hand side of the equation by
- 2, it becomes
- 2k/ -2 = - 18/-2
k = 9
The expression becomes
y = 9x
Therefore, when x = 20,
y = 9 × 20 = 180
Yo sup??
since y varies directly with x we can say
y=kx
at y=-18, x=-2 then
-18=k*(-2)
k=9
therefore
y=9x
at x=20
y=9*20
=180
Hope this helps.
Anna wants to estimate the mean number of siblings for each student in her school. She records the number of siblings for each of 75 randomly selected students in the school. What is the statistic?a. the specific number of siblings for each randomly selected student
b. all the students in the school
c. the mean number of siblings for all students in the school
d. the mean number of siblings for the randomly selected students
e. the 75 randomly selected students
Final answer:
The statistic concerning the number of siblings would be the mean number of siblings for the randomly selected students, not the entire school population.
Explanation:
The question posed by the student pertains to determining the mean number of siblings for students in a school. Anna collected data by randomly selecting 75 students and recording the number of siblings for each student. The statistic in this context is d. the mean number of siblings for the randomly selected students. This is because the statistic refers to a summary measure that is calculated from a sample of data. Therefore, the statistic is the calculated average number of siblings from just those 75 students and not the entire school population.
When discussing sampling methods, one could use a completely random method or use systematic sampling with a tool such as a random number generator for selection. Regardless of the method, the primary criterion is that every member of the population has an equal chance of being included in the sample.
Final answer:
The statistic refers to the mean number of siblings for the randomly selected students, computed by dividing the sum of siblings reported by the 75 students by 75.
Explanation:
The statistic in this scenario is the mean number of siblings for the randomly selected students. The mean, or average, will be calculated by adding up the total number of siblings reported by the 75 students and then dividing that sum by 75. This statistic will serve as an estimate for the mean number of siblings for each student in the whole school. Although option b attempts a similar approach with systematic sampling by selecting every 50th student, the actual calculated mean from the 75 randomly selected students (which is the result of the random sampling method described in option a) is the statistic we are referring to. Option c is asking for a different kind of statistic related to stress scores.
If a chain of 30 identical short springs linked end-to-end has a stiffness of 450 N/m, what is the stiffness of one short spring?
The stiffness of one short spring is 13500 N/m
Solution:
We are given a chain with number of spring N = 30 and linked end to end ( in series) and stiffness of this chain is 450 N/m
We have to find the stiffness of one short spring
The springs are identical, which means they have same stiffness
The stiffness of one spring in series is given as:
[tex]k_i = N \times k_s[/tex]
Where,
N is the number of springs
[tex]k_i[/tex] is the stiffness of one spring
[tex]k_s[/tex] is the stiffness of this chain
Substituting,
[tex]N = 30\\\\K_s = 450[/tex]
Therefore,
[tex]k_i = 30 \times 450\\\\k_i = 13500[/tex]
Thus the stiffness of one short spring is 13500 N/m
If a chain of 30 identical short springs linked end-to-end has a stiffness of 450 N/m , The stiffness of one short spring is 13500 N/m
Given : Stiffness = 450 N/m,
To find : the stiffness of one short spring
According to the given question,
A chain with number of spring N = 30 Chain is linked end to end Stiffness of this chain is 450 N/m
We knows that,
The springs are identical, by which means they have the same stiffness
Hence, The stiffness of one spring will be given as:
[tex]\rm k_i=N \times k_s[/tex]
Where,
N = number of springs [tex]\rm k_i[/tex] = Stiffness of one spring. [tex]\rm k_s[/tex] = Stiffness of given chain.On substituting the values in the formula we will get,
N = 30
[tex]\rm k_s[/tex] = 450
Then,
[tex]\rm k_i =30 \times450\\\\k_i = 13500[/tex]
Therefore, The stiffness of one short spring is 13500 N/m
Learn more about Logical questions here : https://brainly.com/question/14806757
I needz duh help pwease and tank chu?
Answer:
Step-by-step explanation:
Let h represent the number of hours that Jamarcus can rent the truck.
To rent a truck, the charge is $16 per hour and also a fueling fee of $25
The total cost of renting the truck for x hours would be
16h + 25
Since Jamarcus wants to rent the truck and can spend no more than $125, the inequality representing the situation would be
16h + 25 ≤ 125
16h ≤ 125 - 25
16h ≤ 100
h ≤ 6.25
2) 3x - 5 ≥ - 11
3x ≥ - 11 + 5
3x ≥ - 6
x ≥ - 6/3
x ≥ - 2
The correct graph is option A
Answer:
a
Step-by-step explanation:
Given: △ABC; AB=BC, m∠BDA = 60°, BD=4 cm, BD ⊥ BA . Find: DC, AC.
Answer:
DC = 10.93 cm , AC = 9.8 cm
Step-by-step explanation:
From trigonometry;
⇒ Tan 60 = AB/BD
⇒AB = BD Tan 60 ( where BD = 4 cm )
⇒ AB = 6.93 cm
Also, AB=BC , therefore;
⇒ BC = 6.93 cm
⇒ Cos 60 = BD/AD
⇒ AD = BD/ Cos 60 = 4/Cos 60
⇒ AD = 8 cm
From Pythagoras theorem;
⇒ [tex]AC^{2}[/tex] = [tex]AB^{2}[/tex] + [tex]BC^{2}[/tex] = [tex](6.93)^{2}[/tex] + [tex](6.93)^{2}[/tex]
⇒ AC = [tex]\sqrt{96.05}[/tex] = 9.80 cm
⇒ DC = BD + BC = 4 + 6.93
⇒ DC = 10.93 cm
The U.S. Post Office is interested in estimating the mean weight of packages shipped using the overnight service. They plan to sample 300 packages. A pilot sample taken last year showed that the standard deviation in weight was about 0.15 pound. If they are interested in an estimate that has 95 percent confidence, what margin of error can they expect?A. Approximately 0.017 pounds B. About 0.0003 pounds C. About 1.96 D. Can't be determined without knowing the population mean.
Answer: A. Approximately 0.017 pounds
Step-by-step explanation:
Formula to find the margin of error :
[tex]E=z^*\dfrac{s}{\sqrt{n}}[/tex] , where z* = critical value for confidence interval , s= standard deviation , n= sample size.
As per given , we have
s= 0.15 pound
n= 300
Critical value for 95% confidence = 1.96
Then, Margin of error for 9%% confidence interval will be :
[tex]E=(1.96)\dfrac{0.15}{\sqrt{300}}\\\\=0.0169740979142\approx0.017[/tex]
Hence, they can expect a margin error of 0.017 pound (approximately.)
Thus , the correct answer is A. Approximately 0.017 pounds
How do you do this problem?
Answer:
C) ¼ ∫ u⁵ du, where u = sin(4x)
Step-by-step explanation:
∫ cos(4x) sin⁵(4x) dx
Using u substitution:
u = sin(4x)
du = 4 cos(4x) dx
¼ du = cos(4x) dx
¼ ∫ u⁵ du
George's sandbox requires 32 cubic feet of sand to fill how many bags of sand does he need to fill the sand box if each bag holds 2/3 cubic feet of sand
Answer:
48 bags are needed by George to fill his sandbox.
Step-by-step explanation:
Given:
Total capacity of the sandbox (V) = 32 cubic feet
Capacity of each bag = (B) = [tex]\frac{2}{3}[/tex] cubic feet
Now, number of bags required (N) = ?
The formula to find the total number of bags required to fill the sandbox is given as:
[tex]Number\ of\ bags=\frac{Total\ capacity\ of\ sandbox}{Capacity\ of\ each\ bag}\\\\N=\frac{V}{B}[/tex]
Now, plug in the given values and solve for 'N'. This gives,
[tex]N=32\div\frac{2}{3}[/tex]
In order to multiply a whole number with a fraction, we replace the division sing by multiplication and take the reciprocal of the fractional number. This gives,
[tex]N=32\times \frac{3}{2}\\\\N=\frac{32\times 3}{2}\\\\N=16\times 3=48[/tex]
Therefore, 48 bags are needed by George to fill his sandbox.
George needs 48 bags of sand, each holding 2/3 cubic feet, to fill his sandbox that requires 32 cubic feet of sand.
To determine how many bags of sand are required to fill George's sandbox that has a volume of 32 cubic feet, when each bag holds 2/3 cubic feet of sand, we need to perform a division operation. We divide the total volume (32 cubic feet) by the volume each bag holds (2/3 cubic feet).
The calculation is as follows:
Find the inverse of 2/3 which is 3/2.
Multiply the total volume by the inverse. 32 × (3/2) = 32 × 1.5 = 48.
Thus, George would need 48 bags of sand to fill his sandbox.
A wheelchair ramp is to be built from ground level to a platform that is 9 feet above the ground. The angle that the ramp makes with the ground is 11 degrees. What is the length of the ramp?
The answer is in the attachment
Answer: the length if the ramp is 47.17 feet
Step-by-step explanation:
The wheelchair ramp makes an angle of 11 degrees with the ground and forms a right angle triangle. The length of the ramp becomes the hypotenuse of the right angle triangle.
The distance of the platform from the ground level forms the opposite side of the right angle triangle.
To determine the length of the ramp, h, we would apply the Sine trigonometric ratio. It is expressed as
Sin θ = opposite side/hypotenuse
Therefore,
Sin 11 = 9/h
h = 9/Sin 11 = 9/0.1908
h = 47.17 feet
Bill buys a stock that decreases by 20% on the first day, and then on the second day the stock increases by 30% of its value at the end of the first day. What was the overall percent increase in Bill's stock over the two days?
Answer:
4%
Step-by-step explanation:
Let the stock be $10
A 20% decline = 80/100 * 10
= 8
A 30% increase = 30/100 * 8
= 2.4 + 8 = 10.4.
Overall increase = 10.4 - 10 = 0.4
Percentage overall increase = 0.4/10 * 100
= 4%
Ben's car gets between 18 and 21 miles per gallon of gas. If his car's tank holds 15 gallons, what is the range of distance that Ben can drive on one tank of gas?
Answer:
Step-by-step explanation:
Ben's car gets between 18 and 21 miles per gallon of gas. If his car's tank holds 15 gallons, it means that the least distance that Ben can drive on one tank of gas would be
18 × 15 = 270 miles.
Also, the most distance that Ben can drive on one tank of gas would be
21 × 15 = 315 miles.
Therefore the the range of distance that Ben can drive on one tank of gas is between 270 miles and 315 miles. If d represents the distance, then the range is
270 ≤ d ≤ 315)
Anything will help! Thank you.
Step-by-step explanation:
(a) If g is the number of gallons left in the tank, and t is the time in hours since 8AM:
g = 8 gal − (1 gal / 24 mi) (60 mi / 1 hr) (t hr)
g = 8 − 2.5t
(b) If d is the distance traveled:
d = 60 mi/hr × t hr
d = 60t
(c) When George runs out of gas, g = 0.
0 = 8 − 2.5t
t = 3.2
The distance he travels is:
d = 60(3.2)
d = 192
George travels 192 miles.
3.2 hours after 8AM, the time is 11:12AM.
Tyler has two savings accounts that his grandparents opened for him. The two accounts pay 10% and 12% in annual interest; there is $400 more in the account that pays 12% than there is in the other account. If the total interest for a year is $158, how much money does he have in each account?
Answer: the amount of money in the account that earns 10% interest is $500
the amount of money in the account that earns 12% interest is $900
Step-by-step explanation:
Let x represent the amount of money in the account that earns 10% interest.
Let y represent the amount of money in the account that earns 12% interest.
There is $400 more in the account that pays 12% than there is in the other account. This means that
y = x + 400
The formula for determining simple interest is expressed as
I = PRT/100
Where
I represents interest paid on the loan.
P represents the principal or amount taken as loan
R represents interest rate
T represents the duration of the loan in years.
Considering the account that earns 10% interest.
P = x
R = 10
T = 1 year
I = (x × 10 × 1) = 0.1x
Considering the account that earns 12% interest.
P = y
R = 12
T = 1 year
I = (y × 12 × 1) = 0.12x
If the total interest for a year is $158, it means that
0.1x + 0.12y = 158 - - - - - - - - - - -1
Substituting y = x + 400 into equation 1, it becomes
0.1x + 0.12(x + 400) = 158
0.1x + 0.12x + 48 = 158
0.22x = 158 - 48 = 110
x = 110/0.22 = 500
y = x + 400 = 500 + 500
y = 900
Two friends entered a contest jointly and won; however, there is only one prize and it cannot be split. Some methods of selecting who receives the prize are given below. A: Place both names in equal amounts into a hat and draw one without looking. B: Ask a stranger to flip a coin. C: Roll a die and evaluate the outcome as either even or odd. D: Throw a stone closest to an object. E: Play a hand of blackjack. F: Ask a random stranger to select. The methods that are fair include _____ because _____ is flawed due to increased chances of winning based on one's skill, ______ is flawed due to poor randomization, and ______ is flawed due to unequal probabilities of winning and losing.
Answer:
The methods that are fair include __A, B and C__ because __D__ is flawed due to increased chances of winning based on one's skill, ___F___ is flawed due to poor randomization, and ___E___ is flawed due to unequal probabilities of winning and losing.
Step-by-step explanation:
A. Place both names in equal amounts into a hat and draw one without looking:
If you place both names in equal amounts into a hat, then the probability of picking any one of the names would be equal.
For example, if you put 20 pieces of paper containing each name in a hat, then,there will be 40 names in the hat. The probability of picking any one or the other randomly would be:
P = 20/40 =1/2
B. Ask a stranger to flip a coin:
A coin has only two faces, head and tail. Hence, if one person picks head and the other picks tail, the probability of landing on either head or tail is given as:
P = 1/2
C. Roll a die and evaluate the outcome as either even or odd:
A die has 6 faces with equal number of faces with equal number of even numbers and odd numbers, 3. Hence, the probability of rolling an even or odd number is given as:
P = 3/6 = 1/2
D. Throw a stone closest to an object:
This method is not fair because then it depends solely on who can throw the farthest i.e. the physical ability of the throwers.
E. Play a hand of blackjack
This method is flawed because it has unequal probabilities of winning and losing.
F. Ask a random stranger to select:
This method is unfair because it then depends solely on the random picker. The picker could have personal preferences or a bias based on maybe height, beauty, weight, basically anything characteristic. Hence, it is unfair.
Final answer:
Fair methods for deciding who gets the prize include placing both names in a hat, flipping a coin, and rolling a die as they provide equal chances of winning, while throwing a stone, playing blackjack, or asking a stranger can introduce bias or skill, making them unfair.
Explanation:
The methods that are fair include placing both names in a hat, asking a stranger to flip a coin, and rolling a die because these methods provide equal probabilities of winning or losing. Method D: Throwing a stone closest to an object is flawed due to increased chances of winning based on one's skill, which means it's not random. E: Playing a hand of blackjack is flawed due to poor randomization, as the cards dealt can create significantly different chances of winning. F: Asking a random stranger to select is flawed due to unequal probabilities of winning and losing if the stranger holds a bias, even if unintended.
Tossing a coin is a fair way to decide because both outcomes (heads or tails) have equal chances of occurring, which is a 50% chance each way. This is consistent with the principle of randomness and independent events, where the outcomes of past coin tosses do not influence the results of future tosses.
8. Find the inverse of the function.
Y=-3/x+4
Yo sup??
y=-3/x+4
cross multiply
x+4=-3/y
x=-3/y-4
f(y)=-3/y-4
or
f(x)=-3/x-4
=-4x-3/x
The correct answer is option 4
Hope this helps.
Answer:
It is -4x-3/x.
Please look at the questions again! Some of the answers are incorrect. Here is a hint: For question 1, you should not spend more than one-third of $2,000? Calculate: $2,000 / 3. For question 2, how much is the rent and housing expenses? Calculate: $650 + $60 + $10 + $20 + $20. How much can you afford? You should not spend more than one-third of $2,100. Looks like the rent and expenses are too high for your budget, right?
Answer:question 1)$6.66.67
Question2) Rent and expenses =$760
Rent and expenses are too high for the budget
Step-by-step explanation:
1)$2000×(1/3)= $666.67
2)Rent and expenses =$(650+60+10+20+20)= $760
$2100×(1/3)= $700
Rent and expenses should not exceed one third of $2100, which is $700 but it exceeded by$60. Therefore budget is too high.
Final answer:
These Mathematics questions involve calculating budgets to manage income and expenses. For budgeting housing costs, the recommendation is not to exceed one-third of income. A budget table is used to track all monthly expenses against income to determine savings potential and necessary adjustments.
Explanation:
The subject of these questions is Mathematics, specifically focusing on budgeting and personal finance. In these scenarios, students are learning to apply mathematical operations to real-life situations involving income, expenses, and budget planning.
For question 1, you would calculate the maximum amount you should spend on housing from a $2,000 monthly budget by dividing $2,000 by 3, which gives you $666.67. This is because it's recommended to spend no more than one-third of your income on housing.
For question 2, adding together the rent and housing expenses ($650 + $60 + $10 + $20 + $20), you get a total of $760. If you have an income of $2,100, you should not spend more than $700 on housing (which is one-third of your income), so $760 is indeed too high for the budget.
Creating a budget table is an essential skill for financial literacy. When constructing one, you list your monthly income and subtract all your expenses, including housing, utilities, groceries, and any other costs, to see what is left for savings and discretionary spending. For example, with an after-tax monthly income of $2,589.10, if you spend $790 on rent, $75 on a cell phone, and have other listed expenses, you'd subtract all these from your income to see if you can save the desired 10%.
Need help with #1 #4 #7 plz giving 15 points
Answer:
Q1: p = - 33
Q2: d = - 99
Q3: t = - 13
Step-by-step explanation:
Q1: [tex]$ \textbf{-} \frac{\textbf{p}}{\textbf{3}} \hspace{1mm} \textbf{-} \hspace{1mm} \textbf{8} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{3}[/tex]
We solve this taking LCM.
We get: [tex]$ \frac{-p - 24}{3} = 3 $[/tex]
[tex]$ \implies - p - 24 = 9 $[/tex]
[tex]$ \implies \textbf{p} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{- 33} $[/tex]
Q4: [tex]$ \frac{\textbf{d}}{\textbf{11}} \hspace{1mm} \textbf{-} \hspace{1mm} \textbf{4} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{- 13} $[/tex]
Again we proceed like Q1 by taking LCM.
We get: [tex]$ \frac{d - 44}{11} = - 13 $[/tex]
[tex]$ \implies d - 44 = - 13 \times 11 = - 143 $[/tex]
[tex]$ \implies d = - 143 + 44 $[/tex]
[tex]$ \implies \textbf{d} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{- 99} $[/tex]
Q7: 5t + 12 = 4t - 1
We club the like terms on either side.
[tex]$ \implies 5t - 4t = - 1 - 12 $[/tex]
[tex]$ \implies (5 -4)t = - 13 $[/tex]
[tex]$ \implies \textbf{t} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{- 13} $[/tex]
Hence, the answer.
Orange juice, a raisin bagel, and a cup of coffee from Kelly's Koffee Kart cost a total of $3.60. Kelly posts a notice announcing that, effective the following week, the price of orange juice will increase 50% and the price of bagels will increase 20%. After the increase, the same purchase will cost a total of $4.50, and the orange juice will cost twice as much as coffee.
A. Find the price of each item before the increase.
B. What was the cost of a glass of orange juice before the increase?
C. What was the cost of a raisin bagel before the increase?
D. What was the cost of a cup of coffee before the increase?
Answer:
A
Step-by-step explanation:
Parker was able to pay 44% of his college tuition with his scholarship.The remaining $10,054.52 he paid for with a student loan.What was the cost of Parker's tuition?
Answer: the cost of Parker's tuition is $17955
Step-by-step explanation:
Let x represent the cost of Parker's tuition.
Parker was able to pay 44% of his college tuition with his scholarship. This means that the amount that he was able to pay with his scholarship would be
44/100 × x = 0.44 × x = 0.44x
The amount that is remaining for him to pay would be
x - 0.44x = 0.56x
If he paid the remaining $10,054.52 with a student loan, it means that
0.56x = 10054.52
x = 10054.52/0.56
x = 17954.5
Rounding up to the nearest whole number, it becomes
x = $17955
A consumer survey indicates that the average household spends μ =$155 on groceries each week. The distribution of spending amounts is approximately normal with a standard deviation σ =$25. Based on this distribution,
What proportion of the population spends more than $175 per week on groceries?
How much money do you need to spend on groceries each week to be in the top 20% of the distribution?
How much does your family spend per week on groceries, what is your family�s percentile?
Answer:
a) 21.2%
b) $176.05 or more
c) 15.87%
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = $155
Standard Deviation, σ = $25
We are given that the distribution of spending amounts is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
a) P(spends more than $175 per week on groceries)
P(x > 175)
[tex]P( x > 175) = P( z > \displaystyle\frac{175 - 155}{25}) = P(z > 0.8)[/tex]
[tex]= 1 - P(z \leq 0.8)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(x > 175) = 1 - 0.788 = 0.212 = 21.2\%[/tex]
b) P(X > x) = 0.2
We have to find the value of x such that the probability is 0.2
P(X > x)
[tex]P( X > x) = P( z > \displaystyle\frac{x - 155}{25})=0.2[/tex]
[tex]= 1 -P( z \leq \displaystyle\frac{x - 155}{25})=0.2 [/tex]
[tex]=P( z \leq \displaystyle\frac{x - 155}{25})=0.8 [/tex]
Calculation the value from standard normal z table, we have,
[tex]P(z < 0.842) = 0.8[/tex]
[tex]\displaystyle\frac{x - 155}{25} = 0.842\\\\x = 176.05[/tex]
A consumer has to spend approximately $176.05 or greater to be in the top 20% of the distribution.
c) My family spends on average $130 dollars on groceries.
P(less than $130)
[tex]P( x < 130) = P( z < \displaystyle\frac{130 - 155}{25}) = P(z < -1)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(x < 130) = 0.1587 = 15.87\%[/tex]
Thus, my family percentile is 15.87%
a) 21.2%
b) $176.05 or more
c) 15.87%
We are given the following information in the question:
Mean, μ = $155
Standard Deviation, σ = $25
We are given that the distribution of spending amounts is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_(score) = \displaystyle(x-\mu)/(\sigma)[/tex]
a) P(spends more than $175 per week on groceries)
[tex]P(x > 175)P( x > 175) = P( z > \displaystyle(175 - 155)/(25)) = P(z > 0.8)= 1 - P(z \leq 0.8)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(x > 175) = 1 - 0.788 = 0.212 = 21.2\%\\[/tex]
b) [tex]P(X > x) = 0.2[/tex]
We have to find the value of x such that the probability is 0.2
[tex]P(X > x) \\P( X > x) = P( z > \displaystyle(x - 155)/(25))=0.2 \\= 1 -P( z \leq \displaystyle(x - 155)/(25))=0.2 \\=P( z \leq \displaystyle(x - 155)/(25))=0.8[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(z < 0.842) = 0.8\\\displaystyle(x - 155)/(25) = 0.842\n\nx = 176.05[/tex]
A consumer has to spend approximately $176.05 or greater to be in the top 20% of the distribution.
c) My family spends on average $130 dollars on groceries.
P(less than $130)
[tex]P( x < 130) = P( z < \displaystyle(130 - 155)/(25)) = P(z < -1)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(x < 130) = 0.1587 = 15.87\%[/tex]
Thus, my family percentile is 15.87%