How to compare surface area and volume of similar right rectangular prisms.

Question

Right rectangular prism X is similar to right rectangular prism Y. The surface area of right rectangular prism X is 58 square centimeters cm2, and the surface area of right rectangular prism Y is 1,450 cm2. The volume of right rectangular prism Y is 1,250 cubic centimeters cm3. What is the sum of the volumes, in cm3, of right rectangular prism X and right rectangular prism Y?

This question is from an SAT practice test.

Answer 1

Answer 2

CONTEXT: Who Was George Pólya?

George Pólya (1887–1985) was a Hungarian mathematician known for his work in problem-solving and math education. His famous book How to Solve It (1945) introduced a four-step approach:

1. Understand the problem
2. Devise a plan
3. Carry out the plan
4. Look back

This approach guides systematic thinking in mathematics.

PROBLEM:We have two similar rectangular prisms, X and Y:

1. Surface area of Prism X: 58 cm²
2. Surface area of Prism Y: 1,450 cm²
3. Volume of Prism Y: 1,250 cm³

GOAL: Find the sum of the volumes of both prisms, Vx + Vy.

STEP 1: UNDERSTAND THE PROBLEM

Key relationships for similar 3D figures:

1. Surface area: SAx / SAy = k²
2. Volume: Vx / Vy = k³

where k is the scale factor between corresponding sides.

STEP 2: DEVISE A PLAN

1. Use surface areas to find k.
2. Use k to find Vx.
3. Add Vx + Vy to get the total volume.

STEP 3: CARRY OUT THE PLAN

METHOD 1: Step-by-Step

Surface area ratio: SAx / SAy = 58 / 1450 = 0.04

Find scale factor: k² = 0.04 → k = √0.04 = 0.2

Use volume relationship: Vx / Vy = k³ = (0.2)³ = 0.008

Vx = 0.008 × 1,250 = 10 cm³

Sum of volumes: Vx + Vy = 10 + 1,250 = 1,260 cm³

METHOD 2: Symbolic Approach

Express scale factor: k = √(SAx / SAy)

Express volume in terms of k: Vx = Vy × k³ = Vy × [√(SAx / SAy)]³

Sum of volumes formula: Vx + Vy = Vy × [1 + (√(SAx / SAy))³]

Substitute values: Vx + Vy = 1,250 × [1 + (√0.04)³] = 1,250 × [1 + 0.008] = 1,250 × 1.008 = 1,260 cm³

Note: Symbolic manipulation helps conceptual understanding, reduces errors, and creates a reusable formula for similar problems.

STEP 4: LOOK BACK

Verify ratios:

1. SAx / SAy = 0.04 = k² = 0.2² ✓
2. Vx / Vy = 10 / 1,250 = 0.008 = k³ = 0.2³ ✓

Both methods agree: 1,260 cm³

Reasonableness check: Prism X is much smaller, so its volume being a small fraction of Prism Y makes sense.

Answer 3

If you scale an object, length scales by k, area scales by k², volume scales by k³.

1450 = k²·58

1250 = k³·V

Solve for V

Answer 4

Formula you should know:

Surface area of a rectangular prism = 2(length * width + width * height + length * height)

Volume of a rectangular prism = length * width * height

It is given in the problem that both the prisms are similar. Let's say that the scaling factor is x between the prisms.

Area of the larger prism = 2 (x * length * x * width + x * width * x * height + x * length * x * height) = 1450

[after scaling length, width, height by a factor of x]

Factor out x-squared, you'll be left with:

x-squared * 2(length * width + width * height + length * height) = 1450

x-squared times (area of the smaller prism) = 1450

x-squared * 58 = 1450

x-squared = 25

x = 5

That is, the length, width, and height each scale by a factor of 5 if you go from the smaller prism to the larger prism. Or, in other words, if you go from the larger prism to the smaller prism, the scaling factor is 1/5

Volume of the larger prism = 1250 given

x * length * x * width * x * height = 1250

x-cubed * (length * width * height) = 1250

or (length * width * height) = 1250/x-cubed = 1250 / 5 cubed = 10

That is, length * width * height represent the volume of the smaller prism, it is 10

The total volume therefore is 1250 + 10 = 1260 cubic cm

Hope this helps

Answer 5

The two prisms are similar. Therefore the surface areas are in ratio as to the square of the scale.

1450 / 50 = 25

The scale is 5

The volumes are as the cube of the scale, 125

1250 / X = 125, X = 10

Y + X = 1250 + 10

Answer 6