Suppose that water is pouring into a swimming pool in the shape of a right circular cylinder at a constant rate of 7 cubic feet per minute. if the pool has radius 5 feet and height 8 feet, what is the rate of change of the height of the water in the pool when the depth of the water in the pool is 5 feet?
The rate of change of the height of the water in the pool when the depth is 5 feet is approximately 0.089 feet per minute.
Explanation:To find the rate of change of the height of the water in the pool, we can use the formula for the volume of a cylinder.
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
So, when the depth of the water is 5 feet, we can find the rate of change of the height by differentiating the volume equation with respect to time.
Let's calculate it:
V = π(5^2)(h)
dV/dt = π(25)(dh/dt)
Since the volume is increasing at a constant rate of 7 cubic feet per minute, we have dV/dt = 7.
Substituting the given values, we have:
7 = π(25)(dh/dt)
dh/dt = 7/π(25)
So, the rate of change of the height of the water in the pool when the depth is 5 feet is approximately 0.089 feet per minute.
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To find the rate of change of the height of the water in the pool, we can use the concept of related rates. The rate of change is approximately 0.089 cubic feet per minute.
Explanation:To find the rate of change of the height of the water in the pool, we can use the concept of related rates. Let's call the height of the water in the pool 'h' and the rate of change of the height 'dh/dt'. We know that the volume of water in the pool is flowing at a constant rate of 7 cubic feet per minute, therefore the rate of change of the volume of water in the pool is also constant at 7 cubic feet per minute. The volume of a cylinder is given by V = πr^2h, where r is the radius of the pool and h is the height of the water. We can differentiate this equation with respect to time to find the rate of change of the volume.
dV/dt = πr^2(dh/dt)
Since the radius of the pool is constant at 5 feet and the rate of change of the volume is 7 cubic feet per minute, we can substitute these values into the equation and solve for dh/dt.
7 = π(5^2)(dh/dt)
dh/dt = 7/(π(5^2))
Therefore, the rate of change of the height of the water in the pool when the depth of the water is 5 feet is approximately 0.089 cubic feet per minute.
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Suppose that water is stored in a cylindrical tank of radius 5 m. if the height of the water in the tank is h, then the volume of the water is v = Ïr 2h = (25m2)Ïh = 25Ïh m2. if we drain the water at a rate of 250 liters per minute, what is the rate at which the water level inside the tank drops?
To calculate the rate at which the water level drops, use the formula dh/dt = dV/dt / (25Ï) and the given drainage rate of 250 liters per minute. The resulting rate is -0.01/Ï meters per minute, which represents the speed of the water level decreasing.
Explanation:To determine the rate at which the water level inside the cylindrical tank drops, we use the given volume formula of the cylinder, V = 25Ï h, where V is volume in cubic meters, h is height in meters, and Ï is the constant pi (approximately 3.14159).
Draining the tank at a rate of 250 liters per minute corresponds to a volume rate of 0.25 cubic meters per minute (since 1 cubic meter equals 1000 liters). Using the formula dV/dt = 25Ï dh/dt, we want to find the rate dh/dt at which the height of the water decreases.
Rearranging for dh/dt gives us dh/dt = dV/dt / (25Ï). Substituting the known rate of volume change, dV/dt = -0.25 m³/min (negative because the volume is decreasing), we get dh/dt = -0.25 / (25Ï), which simplifies to dh/dt = -0.01 / Ï meters per minute. Therefore, the rate at which the water level drops is -0.01/Ï meters per minute.
Which quadrilateral has two pairs of different but equal adjacent sides?
Simplify. −7i⋅(−8i)
−56
−56i
56i
56
The 6 consecutive integers below add up to 447. x − 2 x − 1 x x + 1 x + 2 x + 3 what is the value of x ? f. 72 g. 73 h. 74 j. 75 k. 76
The value of x is 75.
what is Algebra?A branch of mathematics known as algebra deals with symbols and the mathematical operations performed on them.
Variables are the name given to these symbols because they lack set values.
In order to determine the values, these symbols are also subjected to various addition, subtraction, multiplication, and division arithmetic operations.
Given:
6 consecutive integers are x − 2, x − 1, x, x + 1, x + 2, x + 3
Now, the integers add up to 447.
So, x-2 + x-1 + x+ x +1 + x+2 + x+3= 447
6x -3 + 6 = 447
6x = 450
x= 75
Hence, the value of x is 75.
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the population pf a city is 2,500. if the number of males is 240 more than the number of females how many males and females are there in the city
You need to represent the number of males in terms of females.
Explanation:Since you know the number of males relative to females, it makes sense to represent the number of females as a variable, let's say f.
So then the number of males is f+240 and we know that the number of males plus the number of females is 2500. Knowing this, we can write an equation: f+(f+240)=2500. I put the number of males in brackets there just to make it easy to recognize.
This equation can be condensed into 2f+240=2500 and then solved:
2f=2500−240
f=2500−2402
f=1130
Then, we know the number of females, and we can solve for the number of males from here using our male formula: males=f+240. You should then get 1370 as the number of males.
Checking this answer, we see that 1130 + 1370 does equal 2500.
Subtracting 3xy2 from 8xy2 gives the same result as the expression
A.7xy^2-2xy^2
B.-7xy^2-2xy^2
C.3xy^2-8xy^2
8-3 = 5
7-2 = 5
so the answer would be A
What are the explicit equation and domain for a geometric sequence with a first term of 4 and a second term of -8?
Answer:
The explicit equation for the given geometric sequence is [tex]a_n=4(-2)^{n-1}[/tex]. The domain for the geometric sequence is all positive integers except 0.
Step-by-step explanation:
It is given that the first term of the geometric sequence is 4 and the second term is -8.
[tex]a_1=4,a_2=-8[/tex]
The common ratio for the sequence is
[tex]r=\frac{a_2}{a_1}=\frac{-8}{4}=-2[/tex]
The explicit equation for a given geometric sequence is
[tex]a_n=ar^{n-1}[/tex]
where, a is first term, n is number of term and r is common ratio.
The explicit equation for the given geometric sequence is
[tex]a_n=4(-2)^{n-1}[/tex]
Here n is the number of term. So, the value of n is must be a positive integer except 0.
Therefore the domain for the geometric sequence is all positive integers except 0.
The perimeter, P, of a rectangle is the sum of twice the length and twice the width.
P = 2(l+w) units
P = 2(5)+2(9) units
P = 4x units
P = (l+l)+(w+w) units
P = 2(x+3) units
Answer:
P=2(l+w)
Step-by-step explanation:
It is given that perimeter, P, of a rectangle is the sum of twice the length and twice the width.
Let length of rectangle is l and width of the rectangle is w.
Perimeter = 2 × Length + 2 × Width
[tex]P=2\times l+2\times w[/tex]
[tex]P=2l+2w[/tex]
Taking out common factors.
[tex]P=2(l+w)[/tex]
[Note: (l+l)+(w+w) units is also true but according to the statement only option 1 is true]
Therefore, the correct option is 1.
A piece of string is 325 centimeters long. You need the string to be 1 1/4 meters long. How many centimeters should you cut off
The number of students at Valley View Middle School increased by 10%, to 770 total students. How many students were enrolled at Valley View Middle School before the increase?
There are 700 students were enrolled at Valley View Middle School before the increase.
What is mean by Percentage?
A number or ratio that can be expressed as a fraction of 100 or a relative value indicating hundredth part of any quantity is called percentage.
To Calculate the percent of a number , divide the number by whole number and multiply by 100.
Given that;
The number of students at Valley View Middle School increased by 10%, to 770 total students.
Now,
Since, The number of students at Valley View Middle School increased by 10%,
And, The number student after the increase = 770
So, We can formulate;
The number of students before the increased is calculated as;
= 770 / (1 + 10%)
= 770 / 1.1
= 700
Thus, There are 700 students were enrolled at Valley View Middle School before the increase.
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Let sin a = 12 13 with a in qii and sin b = − 15 17 with b in qiii. find sin(a + b), cos(a + b), and tan(a + b)
The sin(a + b), cos(a + b) and tan(a + b) for angles a and b where sin a = 12/13 and sin b = -15/17 respectively, can be calculated using the formulas for the sine, cosine, and tangent of the sum of two angles. The results are -252/221 for sin(a+b), 56/221 for cos(a+b) and -4.5 for tan(a+b).
Explanation:The given sin values represent the sides of the right triangles in terms of opposite/hypotenuse. Given we are in the second and third quadrants, where cos values are negative, we can use Pythagoras' Theorem, for example, to find cos a = -√(1 - sin²a) = -√(1 - (12/13)²) = -5/13 and analogously, we obtain cos b = -√(1 - sin²b) = 8/17.
Using the formulas for the sine, cosine, and tangent of the sum of two angles:
sin (a + b) = sin a cos b + cos a sin b, we obtain sin(a + b) = 12/13*(-8/17) + 5/13*(-15/17) = -252/221.For cos (a + b) = cos a cos b - sin a sin b, cos(a + b) = -5/13*-8/17 - 12/13*-15/17 = 56/221.And lastly for tan (a + b) = sin (a + b) / cos (a + b), tan(a + b) = -252/221 / 56/221 = -4.5.
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2.Tumford the cats ate 1 5/6 pounds of cat food last week. This week, he ate 3/4 pounds less. How much cat food did he eat this week?
Stan can paint a wall in 40 minutes. if ted works together with stan, it takes both of them 1515 minutes to paint the same wall. how many minutes does it take if ted paints the wall alone?
Simplify (-a2b3)2(c2)0
Isaac read a total of 20 books over 5 months. After belonging to the book club for 7 months, how many books will Isaac have read in all? Assume the relationship is directly proportional.
(easy question i just forgot how to do it lol)
20 books / 5 months = 4 books per month
4 books * 7 months = 28 books total
The system of inequalities and graph the solution on the graph.
True or false: according to the empirical rule, 95% of the data is within three plus or minus standard deviations of the mean.
round to the nearest hundred 5,503,569
Final answer:
To round 5,503,569 to the nearest hundred, observe the hundreds and the following digit. Since the following digit is a 9, round the hundreds digit up to 7 and zeros follow. The rounded number is 5,503,600.
Explanation:
To round the number 5,503,569 to the nearest hundred, we need to look at the digit in the hundreds place and the digit following it. In 5,503,569, the hundreds digit is 6, and the digit to its right is 9. According to rounding rules, if the digit right after the one we are rounding is 5 or greater, we round up. Since 9 is greater than 5, we round the hundreds place up from 6 to 7 and change all the digits to the right of the hundreds place to zero.
The final answer, therefore, is 5,503,600 when 5,503,569 is rounded to the nearest hundred.
Who was a Mexican american who fought for Texan independence from mexico and later became a Texas senator.
Emanuel used the calculations below to find the product of the given fractions. In which step did his error occur?
Answer:
Emanuel made huis mistake in first step and step 1 is incorrect.
Step-by-step explanation:
The given expression is
[tex](\frac{3}{5})(\frac{4}{9})(-\frac{1}{2})[/tex]
It can be written as
[tex]\frac{3}{5})(\frac{4}{9})(\frac{-1}{2}[/tex]
Step 1: [tex](\frac{(3)(4)(-1)}{(5)(9)(2)})[/tex]
Emanuel used negative sign with both numbers 1 and 2. Therefore the first step of Emanuel is incorrect.
Step 2: [tex]\frac{-12}{90}[/tex]
Step 3: [tex]\frac{-2}{15}[/tex]
Therefore Emanuel made huis mistake in first step and step 1 is incorrect.
If p(a|b) = 0.35, p(b) = 0.75 and p(a) = 0.44 are the events a and b independent ?
Final answer:
Two events A and B are considered independent if the probability of A given B (P(A|B)) is equal to the probability of A, and the probability of B given A (P(B|A)) is equal to the probability of B. In this case, events a and b are not independent.
Explanation:
Two events A and B are considered independent if the probability of A given B (P(A|B)) is equal to the probability of A, and the probability of B given A (P(B|A)) is equal to the probability of B.
In this case, if p(a|b) = 0.35, p(b) = 0.75, and p(a) = 0.44, we can determine if events a and b are independent by comparing the given probabilities.
To check if events a and b are independent, we need to find p(a|b) and compare it to p(a), and find p(b|a) and compare it to p(b).
p(a|b) = 0.35 means that the probability of event a occurring given that event b has occurred is 0.35. Since p(a|b) is not equal to p(a), events a and b are not independent.
unit 0 algebra review functions graphing linear equations
How do you factor 8x^2+18x+9=0
A company is creating a box without a top from a piece of cardboard, but cutting out square corners with side length x.
Which expression can be used to determine the greatest possible volume of the cardboard box?
(x−10)(x−30)x
(10−2x)(30−2x)x
(10−x)(30−x)x
(30x−10)(10x−30)
Answer:
(10−2x)(30−2x)x
Step-by-step explanation:
I know how to explain it, but the other person's answer already has a good explanation. I'm just confirming this to be correct! :D
Determine the amount of an investment if 5000 is invested at an interest rate of 4.5% compounded monthly for 10 years. Round your answer to the nearest whole dollar
The amount of an investment of $5,000 at an interest rate of 4.5% compounded monthly for 10 years will be $7,847, rounded to the nearest whole dollar.
Explanation:To determine the amount of an investment that starts at $5,000, with an interest rate of 4.5% compounded monthly for 10 years, we use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the amount of money accumulated after n years, including interest.P is the principal amount (the initial amount of money).r is the annual interest rate (decimal).n is the number of times that interest is compounded per unit t.t is the time the money is invested for in years.Plugging the values into the formula:
A = 5000(1 + 0.045/12)^(12*10)
A = 5000(1 + 0.00375)^(120)
A = 5000(1.00375)^(120)
A = 5000 * 1.569463137
A = $7,847 (rounded to the nearest whole dollar)
The amount of the investment after 10 years, compounded monthly at 4.5% interest, will be $7,847.
Four subtracted from the reciprocal of a number
An equation has solutions of m = –5 and m = 9. Which could be the equation?
Answer:
[tex]y=m^2-4m-45[/tex]
Step-by-step explanation:
An equation has solutions of m = –5 and m = 9
WE are given with the solution. Lets write the solution as factors
When x=a is a solution then factor is (x-a)
[tex]m=-5[/tex] is a solution. change the sign of the solution while writing factor. factor is (m+5)
[tex]m=9[/tex] is a solution, factor is (m-9)
we use the factors to find the equation
[tex]y=(m+5)(m-9)[/tex]
Multiply the factors using FOIL method
[tex]y=m^2-9m+5m-45[/tex]
[tex]y=m^2-4m-45[/tex]
A football team loses 4 yards on each of three plays. Then they complete a pass for 9 yards. What is the change in yardage after this four plays?
Someone please help
What is the vertex of YVT
A) Q
B) V
C) Y
D) T
Answer:
The required vertex of ∠YVT is V
Step-by-step explanation:
To find : Vertex of ∠YVT
Vertex of an angle is defined as the point about which the corresponding angle is formed or measured.
Also, the angle is formed when two rays meet or intersect is each other. so the point of intersection at which the angle is formed is called the vertex of the angle thus formed.
Now, we need to find the vertex of ∠YVT
First check by which two lines the given ∠YVT is formed.
Now, from the diagram ∠YVT is formed by meeting of the lines YV and TV
And the point of meet is V therefore, the angle is formed at V
Hence, The required vertex of ∠YVT is V