A cement truck pours cement into a container in the shape of a cylinder with a radius of 4 feet. The height, h , of the cement in the container increases at a rate of approximately 10 inches per minute. Write an equation that models the volume, V , of the cement in the container in terms of h , then use the equation to determine the time at which the volume is 80π cubic feet. Note that the volume of a cylinder is V=πr2h , where r and h are the radius and height of the cylinder.
The equation that models the volume, V, of the cement in the container in terms of h is:
V = πr^2h
Here, r represents the radius of the cylinder, which is given as 4 feet.
So, the equation becomes:
V = π(4^2)h
V = 16πh
To determine the time at which the volume is 80π cubic feet, we can substitute V = 80π into the equation and solve for h:
80π = 16πh
Dividing both sides by 16π:
5 = h
Therefore, the height of the cement in the container when the volume is 80π cubic feet is 5 feet.
To find the time, we need to use the given rate of change of the height, which is 10 inches per minute. Since the height is given in feet, we need to convert this rate to feet per minute.
There are 12 inches in a foot, so the rate is:
10 inches/minute = 10/12 feet/minute = 5/6 feet/minute
Now we can set up a proportion to find the time it takes for the height to reach 5 feet:
(5/6 feet/minute) : (t minutes) = 5 feet : 1
Cross-multiplying:
5/6 * 1 = 5 * t
5/6 = 5t
Dividing both sides by 5:
1/6 = t
Therefore, it takes t = 1/6 minutes for the volume of the cement in the container to reach 80π cubic feet.
is that corerct?
Apologies for the confusion in the previous response. The equation V = 16πh is incorrect. The correct equation is:
V = πr^2h = π(4^2)h = 16πh
To determine the time at which the volume is 80π cubic feet, we can substitute V = 80π into the equation and solve for h:
80π = 16πh
Dividing both sides by 16π:
5 = h
Therefore, the height of the cement in the container when the volume is 80π cubic feet is 5 feet.
To find the time, we need to use the given rate of change of the height, which is 10 inches per minute. Since the height is given in feet, we need to convert this rate to feet per minute.
There are 12 inches in a foot, so the rate is:
10 inches/minute = 10/12 feet/minute = 5/6 feet/minute
Now we can set up a proportion to find the time it takes for the height to reach 5 feet:
(5/6 feet/minute) : (t minutes) = 5 feet : 1
Cross-multiplying:
5/6 * 1 = 5 * t
5/6 = 5t
Dividing both sides by 5:
1/6 = t
Therefore, it takes t = 1/6 minutes for the volume of the cement in the container to reach 80π cubic feet.
To find an equation that models the volume of the cement in the container in terms of h, we can use the formula for the volume of a cylinder: V = πr^2h.
Given that the radius of the container is 4 feet, we can substitute r = 4 into the formula:
V = π(4^2)h
V = 16πh
Now, to determine the time at which the volume is 80π cubic feet, we can substitute V = 80π into the equation and solve for h:
80π = 16πh
Dividing both sides of the equation by 16π:
5 = h
Therefore, the height (h) of the cement in the container is 5 feet when the volume is 80π cubic feet.
To determine the time, we need to use the information that the height of the cement in the container increases at a rate of approximately 10 inches per minute. Since there are 12 inches in a foot, the rate of increase in feet per minute is 10/12 = 5/6 feet per minute.
To find the time it takes for the height to increase from 0 to 5 feet, we can divide the change in height (5 feet) by the rate of increase (5/6 feet per minute):
t = (5 feet) / (5/6 feet per minute)
t = (5 feet) * (6/5 feet per minute)
t = 6 minutes
Therefore, it takes 6 minutes for the volume of the cement in the container to reach 80π cubic feet.
To write an equation that models the volume of the cement in the container in terms of h, we start with the formula for the volume of a cylinder: V = πr^2h.
Given that the radius of the cylinder is 4 feet, we substitute the value of the radius into the equation: V = π(4^2)h.
This simplifies to: V = 16πh.
Now, we need to determine the time at which the volume is 80π cubic feet. To find this time, we set V equal to 80π and solve for h.
80π = 16πh
Dividing both sides of the equation by 16π, we get:
h = 5
So, the height of the cement in the container when the volume is 80π cubic feet is 5 feet.
Since the height is increasing at a rate of approximately 10 inches per minute, we need to convert this rate to feet per minute.
Since 1 foot is equal to 12 inches, we divide the rate of 10 inches per minute by 12:
10 inches/minute ÷ 12 = 0.8333 feet/minute (rounded to 4 decimal places).
Now, we can determine the time it takes for the volume to reach 80π cubic feet by dividing the change in height (h = 5) by the rate of change (0.8333 feet/minute):
t = h / rate = 5 feet / 0.8333 feet/minute ≈ 5.9999 minutes.
Therefore, at approximately 6 minutes, the volume of the cement in the container will be 80π cubic feet.