Question
A shoe box has a volume of 64 cubic inches and a surface area of 160 square inches. What is the surface area of a similar shoe box if its volume is only 27 cubic inches? Show your work or explain how you got your answer.
Answers
Bot
First, we need to find the dimensions of the original shoe box. We can use the formula for volume of a rectangular prism:
V = lwh
Where V is volume, l is length, w is width, and h is height.
We know that V = 64, so we can plug that in:
64 = lh*w
Next, we can use the formula for surface area of a rectangular prism:
SA = 2lw + 2lh + 2wh
Where SA is surface area.
We know that SA = 160, so we can use the equation we just found for lw:
SA = 2lw + 2lh + 2wh
160 = 2(64/h) + 2lh + 2w(64/lh)
Simplifying:
80/h + 64/l + 64/h = SA
Now we have two equations with two variables (l and h):
64 = lh*w
SA = 80/h + 64/l + 64/h
We can use substitution to solve for l and h in terms of SA:
64 = lh*w
l = 64/(h*w)
h = 64/(l*w)
SA = 80/h + 64/l + 64/h
SA = 80*w*l/(64*w + 64*l*h + 80*h*w) + 64*w*h/(64*w + 64*l*h + 80*h*w) + 64*l*h/(64*w + 64*l*h + 80*h*w)
Now we can plug in V = 27 and use the equation we just found to solve for SA:
27 = lh*w
w = 27/(lh)
SA = 80/h + 64/l + 64/h
SA = 80*l/(27*h) + 64*h/(27*l) + 64*l/(27*h)
Now we plug in the equation we found for l in terms of h and w:
SA = 80*h*w/(27*h) + 64*h/(27*64/(h*w)) + 64*64/(27*h*w)
Simplifying:
SA = 80w + 32h^2 + 128/(27w)
Now we can take the derivative of this equation with respect to w, set it equal to 0 to find the minimum, and solve for w:
dSA/dw = 80 - 128/(27w^2)
0 = 80 - 128/(27w^2)
w^2 = 1280/27
w = sqrt(1280/27)
Now we can plug this value of w back into our equation for SA:
SA = 80w + 32h^2 + 128/(27w)
SA = 80sqrt(1280/27) + 32h^2 + 128/((27/1280)^(1/2))
SA = 320/3 + 32h^2 + (2560/3)^(1/2)
Finally, we can take the derivative of this equation with respect to h, set it equal to 0 to find the minimum, and solve for h:
dSA/dh = 64/h - 80/(27h^2)
0 = 64/h - 80/(27h^2)
h^3 = 256/3
h = (256/3)^(1/3)
Now we plug this value of h back into our equation for SA:
SA = 320/3 + 32((256/3)^(2/3)) + (2560/3)^(1/2)
SA = 320/3 + 256/((27*2^(1/3))
SA = 320/3 + 256/(27*1.25992)
SA = 320/3 + 6.04988
SA = 334.383
Therefore, the surface area of the smaller shoe box is approximately 334.383 square inches.
V = lwh
Where V is volume, l is length, w is width, and h is height.
We know that V = 64, so we can plug that in:
64 = lh*w
Next, we can use the formula for surface area of a rectangular prism:
SA = 2lw + 2lh + 2wh
Where SA is surface area.
We know that SA = 160, so we can use the equation we just found for lw:
SA = 2lw + 2lh + 2wh
160 = 2(64/h) + 2lh + 2w(64/lh)
Simplifying:
80/h + 64/l + 64/h = SA
Now we have two equations with two variables (l and h):
64 = lh*w
SA = 80/h + 64/l + 64/h
We can use substitution to solve for l and h in terms of SA:
64 = lh*w
l = 64/(h*w)
h = 64/(l*w)
SA = 80/h + 64/l + 64/h
SA = 80*w*l/(64*w + 64*l*h + 80*h*w) + 64*w*h/(64*w + 64*l*h + 80*h*w) + 64*l*h/(64*w + 64*l*h + 80*h*w)
Now we can plug in V = 27 and use the equation we just found to solve for SA:
27 = lh*w
w = 27/(lh)
SA = 80/h + 64/l + 64/h
SA = 80*l/(27*h) + 64*h/(27*l) + 64*l/(27*h)
Now we plug in the equation we found for l in terms of h and w:
SA = 80*h*w/(27*h) + 64*h/(27*64/(h*w)) + 64*64/(27*h*w)
Simplifying:
SA = 80w + 32h^2 + 128/(27w)
Now we can take the derivative of this equation with respect to w, set it equal to 0 to find the minimum, and solve for w:
dSA/dw = 80 - 128/(27w^2)
0 = 80 - 128/(27w^2)
w^2 = 1280/27
w = sqrt(1280/27)
Now we can plug this value of w back into our equation for SA:
SA = 80w + 32h^2 + 128/(27w)
SA = 80sqrt(1280/27) + 32h^2 + 128/((27/1280)^(1/2))
SA = 320/3 + 32h^2 + (2560/3)^(1/2)
Finally, we can take the derivative of this equation with respect to h, set it equal to 0 to find the minimum, and solve for h:
dSA/dh = 64/h - 80/(27h^2)
0 = 64/h - 80/(27h^2)
h^3 = 256/3
h = (256/3)^(1/3)
Now we plug this value of h back into our equation for SA:
SA = 320/3 + 32((256/3)^(2/3)) + (2560/3)^(1/2)
SA = 320/3 + 256/((27*2^(1/3))
SA = 320/3 + 256/(27*1.25992)
SA = 320/3 + 6.04988
SA = 334.383
Therefore, the surface area of the smaller shoe box is approximately 334.383 square inches.
oobleck
AAAaannndd the bot gets it wrong yet again!
What a lot of nonsense and gobbledegook!
the volumes are in the ratio 27/64 = (3/4)^3
so the surface areas are in the ratio (3/4)^2 = 9/16
9/16 * 160 in^2 = 90 in^2
What a lot of nonsense and gobbledegook!
the volumes are in the ratio 27/64 = (3/4)^3
so the surface areas are in the ratio (3/4)^2 = 9/16
9/16 * 160 in^2 = 90 in^2
Bot
I apologize for the mistake in my previous response. You are correct that the volumes are in the ratio (3/4)^3 and therefore the surface areas are in the ratio (3/4)^2. Multiplying this ratio by the original surface area of 160 square inches gives us:
(3/4)^2 * 160 = 90 square inches
Therefore, the surface area of the similar shoe box with a volume of 27 cubic inches is 90 square inches. Thank you for correcting me.
(3/4)^2 * 160 = 90 square inches
Therefore, the surface area of the similar shoe box with a volume of 27 cubic inches is 90 square inches. Thank you for correcting me.