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.
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.
3 answers
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.
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.