A, B and C are three similar shapes. The surface area of shape A is 4 cm2
.
The surface area of shape B is 25 cm2
.
The ratio of the volume of shape B to the volume of shape C is 27 : 64.
Work out the ratio of the height of shape A to the height of shape C.
Simplify your answer, and give it in the form A : C.

+explanation pleasee :)

Explanation

Shape A has surface area 4 cm^2

Shape B has surface area 25 cm^2

The ratio of the surface areas, from A to B, is 4:25

Apply the square root to both parts and we get 2:5 as the ratio of the corresponding sides. If shape A has a height of 2, then shape B will have a corresponding height of 5. I recommend drawing out two squares with these properties to see that their areas form the ratio mentioned above.

The ratio of the volume of shape B to shape C's volume is 27:64. Apply the cube root to both parts to get 3:4. This means if the height of shape B is 3, then the corresponding height on shape C is 4.

We found these ratios for the sides

The goal is to find the ratio A:B:C which ties all three shapes together showing how they all connect in terms of their corresponding sides.

Note how in A:B = 2:5 we have B = 5, but in B:C = 3:4 we have B = 3. We cannot combine the two ratios in this form since the B terms don't match.

The good news is that we can apply scaling to get the B values to match up

The ratio 2:5 is equivalent to 6:15 after multiplying both parts by 3

The ratio 3:4 is equivalent to 15:20 after multiplying both parts by 5.

The multipliers 3 and 5 are used because note how 15 is the LCM of those previous B values mentioned.

After scaling those ratios, we have

From here, we can quickly see that A:B:C = 6:15:20. This is possible because the '15's match up for the B terms.

If shape A has a height of 6, then shape B has a height of 15, and C has a height of 20. All of these heights correspond to one another.

Then we just erase out the B part to conclude A:C = 6:20 which reduces to A:C = 3:10 after dividing both parts by 2.

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