A candy company needs a custom box for their truffles. The box they've chose is in the shape of a cylinder with a hemisphere of the same radius on top. The total volume of the box is V= (1/2) ((4pir^2)/(3))+ pir^2 (y-r), where y is the height of the box and r is the radius of the box. Originally, the candy box was designed to have a height of 6 inches and a radius of 2 in, but the shipper suggests that the boxes be made slightly shorter. You now need to adjust the radius so that the height is reduced to 5.75 in. but the volume remains constant.
A.Find the value of dr/dy at the point r=2, y=6.
B.Use your value of dr/dy to approximate the new radius for these boxes.
the answer is 2.05 inches
To find the value of dr/dy at the point r=2, y=6, we need to differentiate the volume equation V with respect to r and y.
Given: V = (1/2) ((4πr^2)/(3)) + πr^2 (y-r)
Step 1: Differentiating the volume equation V with respect to r:
dV/dr = (1/2)(8πr/3) + π(2r)(y-r) - πr^2
Simplifying the equation:
dV/dr = 4πr/3 + 2πy - 3πr - πr^2
Step 2: Differentiating the volume equation V with respect to y:
dV/dy = πr^2 - πr^2
Simplifying the equation gives:
dV/dy = 0
The value of dy/dr at the point r=2, y=6 can be found by substituting these values into the equations we derived.
dV/dr = 4π(2)/3 + 2π(6) - 3π(2) - π(2)^2
Simplifying further:
dV/dr = (8π/3) + 12π - 12π - 4π
dV/dr = 8π/3
So, the value of dr/dy at the point r=2, y=6 is 8π/3.
To approximate the new radius for these boxes when the height is reduced to 5.75 inches while keeping the volume constant, we can use the following formula:
dr = (dr/dy) * (dy)
Substituting the value of dr/dy, which we found earlier, dr/dy = 8π/3, and the change in height dy as -0.25 (since the height is reduced), we can calculate the change in radius dr:
dr = (8π/3) * (-0.25)
dr = (-2π/3)
To find the new radius, subtract the change in radius from the original radius:
New Radius = 2 - (2π/3)
Therefore, the new radius for these boxes is approximately 0.535 inches when the height is reduced to 5.75 inches, while keeping the volume constant.
To find the value of dr/dy at the point r=2, y=6, we need to take the derivative of the volume equation with respect to both r and y, and then evaluate it at the given point.
Let's start by finding the derivative ∂V/∂r:
V = (1/2) ((4πr^2)/(3)) + πr^2 (y-r)
Take the partial derivative with respect to r:
∂V/∂r = ∂(1/2) ((4πr^2)/(3))/∂r + ∂(πr^2 (y-r))/∂r
= (8πr/3) - 2πr(y-r)
= (8πr/3) - 2πry + 2πr^2
Now, let's find the derivative ∂V/∂y:
∂V/∂y = ∂(1/2) ((4πr^2)/(3))/∂y + ∂(πr^2 (y-r))/∂y
= 0 + πr^2
Evaluate the derivatives at r=2 and y=6:
∂V/∂r at r=2, y=6:
(8π(2)/3) - 2π(2)(6) + 2π(2)^2
= (16π/3) - 24π + 8π
= -24π + 8π
= -16π
∂V/∂y at r=2, y=6:
π(2)^2
= π(4)
= 4π
Now we can find the value of dr/dy by dividing ∂V/∂r with ∂V/∂y at r=2, y=6:
dr/dy = (∂V/∂r)/(∂V/∂y)
dr/dy = (-16π)/(4π)
dr/dy = -4
The value of dr/dy at the point r=2, y=6 is -4.
To approximate the new radius for these boxes, we need to use the value of dr/dy to adjust the original radius.
Original radius: r = 2 inches
Original height: y = 6 inches
New height: y = 5.75 inches
To keep the volume constant, we need to find the new radius r'.
Using the derivative equation:
dr/dy = (∂V/∂r)/(∂V/∂y) = -4
We can rewrite this as:
-4 = (∂V/∂r)/(∂V/∂y)
Substituting the values of ∂V/∂r and ∂V/∂y at r=2, y=6:
-4 = (-16π)/(4π)
Simplifying:
-4 = -4
This indicates that the new radius should be the same as the original radius, r' = r.
So, the new radius for these boxes will also be 2 inches.
The formula has a typo. Should be
V= (1/2) ((4pir^3)/(3))+ pi r^2 (y-r)
or
v = 2π/3 r^3 + πr^2 (y-r)
= πr^2 y - π/3 r^3
2πryr' + πr^2 - πr^2 r' = 0
r' = r/(r-2y)
r'(6) = 2/(2-12) = -1/5
dr = dr/dy * dy
= -1/5 (-.25) = 0.05
so, r = 2.05