To write a two-variable equation, I would first need to know how much Maya’s allowance was. Then, I would need the cost of playing the arcade game and of riding the Ferris wheel. I could let the equation be cost of playing the arcade games plus cost of riding the Ferris wheel equals the total allowance. My variables would represent the number of times Maya played the arcade game and the number of times she rode the Ferris wheel. With this equation I could solve for how many times she rode the Ferris wheel given the number of times she played the arcade game u welcome
The width of a rectangle is 9 units less than it length L. Write an expression that shows how wide the rectangle is in terms of L.
12. A race car is running practice laps in preparation for an upcoming race. To judge how
the car is performing, the crew takes measurements of the car’s speed S(t) (in miles per
hour, or mph) every minute. The measurements are given in the table below.
t (minutes) S(t) (mph)
0 201
1 205
2 208
3 214
4 218
5 212
6 219
7 223
8 220
9 221
10 217
11 218
A. Use the trapezoid rule with 4 equal subdivisions to approximate the total distance the car
traveled (in miles) over the first 12 minutes.
B.Find one approximation for Sv(6), including the units. Explain what this quantity means in
the context of the problem.
C. What was the car’s average speed in mph over the first 12 minutes? If the car needs to
have an average speed of 210 mph to qualify for the race, is it currently running fast enough
to qualify?
The student's problem involves the use of the trapezoid rule to approximate total distance traveled by a race car, finding the speed at a particular time point (Sv(6)), and calculating the car's average speed to see if it meets the qualifying speed.
Explanation:To solve this problem, we need to conduct several mathematical operations. Firstly, let's use the trapezoid rule to approximate the total distance the car traveled in 12 minutes. Then we'll find an approximation for Sv(6) and explain its meaning. Finally, we'll calculate the car's average speed and determine if it's fast enough to qualify for the race.
For the trapezoid rule, remember that it's structured as (b-a)/2n [f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn) + f(xn+1)]. We'll create 4 equal subdivisions over the first 12 minutes. Due to the lack of full data set, let's suppose the missing speeds are similar to the closely related ones provided.Sv(6) stands for the velocity, or speed, at the 6th minute, which is 219 mph. This represents how fast the car was going at that specific moment.To get the average speed, add up all the speeds given and divide by the number of measurements. If the resultant speed is at least 210 mph, then the car is fast enough to qualify for the race.Learn more about Trapezoid Rule & Average Speed here:https://brainly.com/question/33247395
#SPJ3
Write the ratio using fraction notation and reduce.
20 minutes to 1 hour
(a) 20
(b )1/3
(c) 1/20
(d) 3
Calculus: Help ASAP
Evaluate exactly the value of the integral from negative 1 to 0 of the product of the cube of the quantity 4 times x to the 6th power plus 2 times x and 12 times x to the 5th power plus 1, dx. Your work must include the use of substitution and the antiderivative.
Answer:
2.264 (3 d.p.)
Step-by-step explanation:
Given integral:
[tex]\displaystyle \int^0_{-1} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x[/tex]
First, evaluate the indefinite integral using the method of substitution.
[tex]\textsf{Let} \;\;u = 4x^6+2x[/tex]
Find du/dx and rewrite it so that dx is on its own:
[tex]\dfrac{\text{d}u}{\text{d}x}=24x^5+2 \implies \text{d}x=\dfrac{1}{24x^5+2}\; \text{d}u[/tex]
Rewrite the original integral in terms of u and du, and evaluate:
[tex]\begin{aligned}\displaystyle \int\left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x&=\int \left(u\right)^3\left(12x^5+1\right)\cdot \dfrac{1}{24x^5+2}\; \text{d}u\\\\&=\int \left(u\right)^3\left(12x^5+1\right)\cdot \dfrac{1}{2(12x^5+1)}\; \text{d}u\\\\&=\int \dfrac{u^3\left(12x^5+1\right)}{2(12x^5+1)}\; \text{d}u\\\\&=\displaystyle \int \dfrac{u^3}{2}\; \text{d}u\\\\&=\dfrac{u^{3+1}}{2(3+1)}+C\\\\&=\dfrac{u^4}{8}+C\end{aligned}[/tex]
Substitute back u = 4x⁶ + 2x:
[tex]=\dfrac{(4x^6+2x)^4}{8}+C[/tex]
Therefore:
[tex]\displaystyle \int \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x=\dfrac{(4x^6+2x)^4}{8}+C[/tex]
To evaluate the definite integral, we must first determine any intervals within the given interval -1 ≤ x ≤ 0 where the curve lies below the x-axis. This is because when we integrate a function that lies below the x-axis, it will give a negative area value.
Find the x-intercepts by setting the function to zero and solving for x.
[tex]\left(4x^6+2x\right)^3\left(12x^5+1\right)=0[/tex]
Therefore:
[tex]\begin{aligned}\left(4x^6+2x\right)^3&=0\\4x^6+2x&=0\\x(4x^5+2)&=0\end{aligned}[/tex]
[tex]x=0[/tex]
[tex]\begin{aligned}4x^5+2&=0\\4x^5&=-2\\x^5&=-\frac{1}{2}\\x&=\sqrt[5]{-\dfrac{1}{2}}\end{aligned}[/tex]
[tex]\begin{aligned}12x^5+1&=0\\12x^5&=-1\\x^5&=-\dfrac{1}{12}\\x&=\sqrt[5]{-\dfrac{1}{12}}\end{aligned}[/tex]
Therefore, the curve of the function is:
Below the x-axis between -1 and ⁵√(-1/2).Above the x-axis between ⁵√(-1/2) and ⁵√(-1/12).Below the x-axis between ⁵√(-1/12) and 0.So to calculate the total area, we need to calculate the positive and negative areas separately and then add them together, remembering that if you integrate a function to find an area that lies below the x-axis, it will give a negative value.
Integrate the function between -1 and ⁵√(-1/2).
As the area is below the x-axis, we need to negate the integral so that the resulting area is positive:
[tex]\begin{aligned}A_1&=-\displaystyle \int_{-1}^{\sqrt[5]{-\frac{1}{2}}} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x\\\\&=-\left[\dfrac{(4x^6+2x)^4}{8}\right]_{-1}^{\sqrt[5]{-\frac{1}{2}}}\\\\&=-\left[\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{2}}\right)^6+2\left(\sqrt[5]{-\frac{1}{2}}\right)\right)^4}{8}\right)-\left(\dfrac{(4(-1)^6+2(-1))^4}{8}\right)\right]\\\\&=-[0-2]\\\\&=2\end{aligned}[/tex]
Integrate the function between ⁵√(-1/2) and ⁵√(-1/12).
[tex]\begin{aligned}A_2&=\displaystyle \int_{\sqrt[5]{-\frac{1}{2}}} ^{\sqrt[5]{-\frac{1}{12}}} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x\\\\&=\left[\dfrac{(4x^6+2x)^4}{8}\right]_{\sqrt[5]{-\frac{1}{2}}}^{\sqrt[5]{-\frac{1}{12}}}\\\\&=\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{12}}\right)^6+2\left(\sqrt[5]{-\frac{1}{12}}\right)\right)^4}{8}\right)-\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{2}}\right)^6+2\left(\sqrt[5]{-\frac{1}{2}}\right)\right)^4}{8}\right)\\\\\end{aligned}[/tex]
[tex]\begin{aligned}&=\dfrac{625}{648\sqrt[5]{12^4}}-0\\\\&=0.132117398...\end{aligned}[/tex]
Integrate the function between ⁵√(-1/12) and 0.
As the area is below the x-axis, we need to negate the integral so that the resulting area is positive:
[tex]\begin{aligned}A_3&=-\displaystyle \int_{\sqrt[5]{-\frac{1}{12}}}^0 \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x\\\\&=-\left[\dfrac{(4x^6+2x)^4}{8}\right]_{\sqrt[5]{-\frac{1}{12}}}^0\\\\&=-\left[\left(\dfrac{(4(0)^6+2(0))^4}{8}\right)-\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{12}}\right)^6+2\left(\sqrt[5]{-\frac{1}{12}}\right)\right)^4}{8}\right)\right]\\\\&=-\left[0-\dfrac{625}{648\sqrt[5]{12^4}}\right]\\\\&=\dfrac{625}{648\sqrt[5]{12^4}}\\\\&=0.132117398...\\\\\end{aligned}[/tex]
To evaluate the definite integral, sum A₁, A₂ and A₃:
[tex]\begin{aligned}\displaystyle \int^0_{-1} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x&=2+2\left( \dfrac{625}{648\sqrt[5]{12^4}}\right)\\\\&=2+ \dfrac{625}{324\sqrt[5]{12^4}}\right}\\\\&=2.264\; \sf (3\;d.p.)\end{aligned}[/tex]
One machine can make 138,800 small paper clips in one day.Another machine can make 84,250 large paper clips in one day.How many more small paper clips than large paper clips are made by two machines in one day
What is the perimeter of the triangle shown on the coordinate plane, to the nearest tenth of a unit?
a.14.6
b.15.5
c.21.0
d.21.6
Combine like terms.
9 + 3x – 9x + 16
A.19x
B.3x + 16
C.25 – 6x
D.18x + 3x + 16
measures of an angle and five times it’s complement is 298 what is the measure of the angle
PLZ HELP!!!! WILL GIVE BRAINLIEST ANSWER!!!
△E′F′G′ is the image of △EFG under a dilation through point C.
What scale factor was used in the dilation?
the correct answer is - 1/2
Answer: The value of the scale factor is [tex]-\frac{1}{2}[/tex].
Explanation:
It is given that the △E′F′G′ is the image of △EFG under a dilation through point C.
We know that the length of sides of image are in the proportion of length of sides of preimage. That proportion factor is also known as scale factor.
It is given that the length of FG is 8 cm and the length of F'G' is 4 cm.
[tex]k=\frac{F'G'}{FG} =\frac{4}{8} =\frac{1}{2}[/tex]
Therefore the side of image and preimage are in proportion of [tex]\frac{1}{2}[/tex].
Since the image and preimage are not on the same side of the center of dilation it means the factor must be negative. So,
[tex]k=-\frac{1}{2}[/tex].
Therefore, [tex]-\frac{1}{2}[/tex] is the scale factor of dilation.
Zoe is taking the measurements of her lawn. her lawn is 64.15 feet long and 22.70 meters wide. one foot is equivalent to 0.3048 meters. which dimension of zoe's lawn is longer, and how many yards longer is it?
Answer:
width of the lawn is greater than the length of the lawn.Step-by-step explanation:
The length of her lawn=64.15 feet.
Width of her lawn=22.70 meters.
we are given that:
1 foot=0.3048 meters.
Hence,
64.15 feet=21.86232 meters.
Hence, length of lawn=21.86232 meters.
Hence, width of the lawn is greater than the length of the lawn.
( since 22.70 meters> 21.862332 meters)
Also 1 meter=1.0936 yards.
so, width of lawn in yards is:
22.70×1.0936=24.82472 yards.
length of lawn is: 23.9086462752 yards.
Hence, width of lawn is 0.9160737 yards more than the length of lawn.
Alvin's age is three times elga's age. the sum of their ages is 32 . what is elga's age
Final answer:
Elga's age is determined by setting up an equation E + 3E = 32 and solving for E. After simplifying, we find that Elga is 8 years old.
Explanation:
To solve this problem, we can use algebra to set up two equations based on the information given that Alvin's age is three times Elga's age and the sum of their ages is 32.
Let's let E represent Elga's age. According to the problem, Alvin's age will be 3E because it is three times Elga's age. We can write the following equation to represent the sum of their ages:
E + 3E = 32
Combining like terms, we have:
4E = 32
Dividing both sides by 4 to solve for E, we get:
E = 32 / 4
Therefore, Elga's age is:
E = 8
Elga is 8 years old.
which of the following equations represents a proportional relationship? Choose all that apply.
A. x=2y
B.a = 1/3b
C.4x=y-2
D.7/3x=n
E.1/2=1/4
The options A, B, and D have a proportional relationship.
What is a proportional relationship?It is the relationship between two variables where their ratios are equivalent.
Consider the first equation, x = 2y.
It can be written like the ratio of x and y.
i.e [tex]\frac{x}{y} -\frac{2}{1}[/tex]
So, option A is proportional at x = 2 and y = 1.
For option B, It can rewrite as
[tex]\frac{a}{b} =\frac{1}{3}[/tex]
So, option B is proportional when a = 1 and b = 3.
Option C is not proportional since it can't be written as a ratio.
Option D can be written as
[tex]\frac{7}{3} =\frac{n}{x}[/tex]
It is proportional when n = 7 and x = 3.
Option E is not proportional since it is not equal.
Therefore the proportional options are A,B, and D.
To learn more about proportional relationship, use the link given below:
https://brainly.com/question/29765554
#SPJ2
Rashawn read 25 pages of his book each day until he finished the book. His book was 400 pages long.
Which sketch represents this situation?
If PR = 4x – 2 and RS = 3x – 5, which expression represents PS? x – 7 x – 3 7x – 7 7x + 3
we have that
[tex]PR = 4x - 2\\RS = 3x - 5[/tex]
we know that
[tex]PS=PR+RS[/tex]
substitute the values of PR and RS in the formula above
[tex]PS=(4x-2)+(3x-5)[/tex]
Combine like terms
[tex]PS=(7x-7)[/tex]
therefore
the answer is
[tex]PS=(7x-7)[/tex]
The expression which represents PS is; 7x -7
Addition of algebraic expressionsAccording to the question;
The segment, PR = 4x – 2 and RS = 3x – 5.We are required to find; PS.From observation;
PS = PR + RSPS = 4x -2 + 3x -5PS = 4x + 3x -2 -5PS = 7x -7.Read more on addition of algebraic expressions;
https://brainly.com/question/2164351
Write an equation that represents "the difference between a number and eight is twenty-two". 8 - N = 22 N - 8 = 22 8 - 22N = 22
Answer:
N - 8 = 22
Step-by-step explanation:
A local charity sponsors a 5K race to raise money. It receives $25 per race entry and $5,000 in donations, but it must spend $5 per race entry to cover the cost of the race. Write and solve an inequality to determine the number of race entries the charity needs to raise more than $25,000. Show your work!
Question 1 options:
Spell check
they receive 25 per entry but 5 goes towards the cost of the race
so 25-5 = 20 per entry is for the charity
so 20x + 5000 = 25000
subtract 5000 from each side
20x= 20000
divide both sides by 20
x = 20000/ 20
x = 1000
they need at least 1000 entries to make 25000
so to make more than 25000, they need 1001 entries
HELPS PLS ryad borrowed $1450 and made 18 payment of $95.25 how much did he pay in interest?
A spherical balloon is inflated so that its volume is increasing at the rate of 3.1 {\rm ft}^3{\rm /min}. how rapidly is the diameter of the balloon increasing when the diameter is 1.6 feet?
What does it mean if two angles are congruent?
1/4=1/4h + 4
I understand how to do the 2-step equations, but not the ones with 2 fractions.
You decide to play roulette 200 times each time betting the same amount on red. you will loe money if you win on fewer than 100 of the plays. what is the probability that you will lose money? ( check that the normal approximation is permissible and use it to find this probability) in general, if you bet the same.
The probability of losing money in this scenario is 23.89%.
Probability of Losing Money in Roulette (200 spins, red bet)
Step-by-Step Calculation:
1) Probability of winning on one spin:
Red covers 18 out of 38 numbers (European roulette).
P(win) = 18/38 ≈ 0.474.
2) Number of wins needed to break even:
Divide total spins by 2 due to 1:1 payout on red.
200 spins / 2 = 100 wins needed.
3) Binomial distribution:
200 trials (spins), each with 2 outcomes (win/loss).
Success probability = P(win) = 0.474.
Failure probability = P(loss) = 1 - P(win) = 0.526.
4) Normal approximation:
Check conditions for normal approximation:
np > 5 (200 × 0.474 ≈ 95 > 5)
nq > 5 (200 ×0.526 ≈ 105 > 5)
Both conditions met, so normal approximation is valid.
5) Calculate mean and standard deviation:
Mean (μ) = np = 200 × 0.474 ≈ 94.8.
Standard deviation (σ) = sqrt(npq) = sqrt(200 * 0.474 * 0.526) ≈ 7.4.
6) Find probability of losing money:
We need wins < 100, which translates to losses ≥ 100.
Convert losses to z-scores:
z = (x - μ) / σ = (100 - 94.8) / 7.4 ≈ 0.71.
Look up probability of z ≥ 0.71 in standard normal distribution table:
P(z ≥ 0.71) ≈ 0.2389.
7) Generalization:
For any number of spins (n) and break-even win threshold (k), the steps are similar. Use the binomial distribution or its normal approximation to calculate the probability of winning k or fewer times.
The length of a rectangular garden is 3yd more than twice it’s width. The perimeter of the garden is 36yd. What are the width and length of the garden?
A package is in the shape of a triangular prism. The bases are right triangles with perpendicular legs measuring 9 centimeters and 12 centimeters. The distance between the bases is 10 centimeters. What is the surface area of the triangular prism? square centimeters
The surface area of the triangular prism is 468 (square cm).
The surface area A of a triangular prism is given by the formula:
[tex]\[ A = 2A_{\text{base}} + P_{\text{base}} \times h \][/tex]
where:
- [tex]\( A_{\text{base}} \)[/tex] is the area of one of the triangular bases,
- [tex]\( P_{\text{base}} \)[/tex] is the perimeter of one of the triangular bases,
- h is the distance between the bases.
The area of a right triangle is given by:
[tex]\[ A_{\text{base}} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
And the perimeter of a right triangle is the sum of the lengths of its three sides.
Given that the legs of the right triangles forming the bases are 9 cm and 12 cm, and the distance between the bases is 10 cm, we can calculate the surface area.
1. Area of the triangular base [tex](\(A_{\text{base}})\)[/tex]:
[tex]\[ A_{\text{base}} = \frac{1}{2} \times 9 \times 12 \][/tex]
2. Perimeter of the triangular base [tex](\(P_{\text{base}})\)[/tex]:
[tex]\[ P_{\text{base}} = 9 + 12 + \sqrt{9^2 + 12^2} \][/tex]
3. Surface area of the triangular prism A:
[tex]\[ A = 2 \times A_{\text{base}} + P_{\text{base}} \times 10 \][/tex]
Calculate each part and find the total surface area.
[tex]\[ A_{\text{base}} = \frac{1}{2} \times 9 \times 12 = 54 \, \text{cm}^2 \][/tex]
[tex]\[ P_{\text{base}} = 9 + 12 + \sqrt{9^2 + 12^2} = 9 + 12 + 15 = 36 \, \text{cm} \][/tex]
[tex]\[ A = 2 \times 54 + 36 \times 10 = 108 + 360 = 468 \, \text{cm}^2 \][/tex]
So, the surface area of the triangular prism is [tex]\(468 \, \text{cm}^2\)[/tex].
Please someone help
Point G is between points F and H. FH = 102, FG = 5x + 9, and HG = 9x − 5. Show your work.
A.) What is the value of x ?
B.) What is the length of ̅̅̅̅FG?
C.) What is HG?
How many lines of symmetry does a regular polygon with 12 sides have?
What is the value of x? (7x-8) (6x+11)
we know that
Vertical angles are a pair of opposite and congruent angles formed by intersecting lines
In this problem
[tex](7x-8)=(6x+11)[/tex] --------> by vertical angles
Solve for x
Combine like terms
[tex](7x-6x)=(11+8)[/tex]
[tex]x=19\ degrees[/tex]
therefore
the answer is
the value of x is [tex]19\ degrees[/tex]
The measures of the 3 sides of a triangle can be represented by algebraic expressions x, 3x-1, and 4x＋2.the perimeter of the triangle is 81inches. What are the lengths of the sides of the triangle?
what is the measure on angle A?
- 110
-70
250
55
Does a regular hexagon have 2 parallel sides