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.

Answer 1

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.

Answer 2

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

Answer 3

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.