Applied math- Scale

Question

A pet store designs fish tanks that are in the shape of a rectangular prism. The small tank has a capacity of 1920 cu in. The scale factor that relates the dimensions of the small tank to the dimensions of the medium tank are 1.25. The scale factor that relates the dimensions of the medium tank to the dimensions of the large tank are 1.2.

Glass costs the store $12 to put in the small tank.

Find the capacity of the medium and large tanks and find the cost of the glass for the medium and large tanks.

1. 1 mark for finding the surface area scale factor for the medium tank
2. 1 mark for finding the surface area scale factor for the large tank
3. 1 mark for finding the cost of glass for the medium & large tank
4. 1 mark for finding the volume scale factor for the medium tank
5. 1 mark for finding the volume scale factor for the large tank
6. 1 mark for finding the volume of the medium & large tank

Answer 1

Anything with CUBIC UNITS is a volume, and the volume of a rectangular prism is V = L W H = 1920 cu inches this is the volume for the small tank.

"DIMENSIONS" SCALE FACTOR RELATES "LINEAR" QUANTITIES. feet:feet, meters:meters

Small:Med = 1.25

Med:Large = 1.2

Small:Large = 1.25 * 1.2 = 1.5

So if the Length of the SMALL is 16 inches, then the length of the MEDIUM is 16 * 1.25 = 20 inches

If the length of the MEDIUM is 20 inches, the the length of the LARGE is 20 * 1.2 = 24 inches.

Check: Small * 1.5 should = Large length = 16 * 1.5 = 24 inches √

If we want the Surface Area scale factor, we must square the Dimensions scale factor (b/c area has SQUARE units) So for the Surface Area:

Small:Med = 1.25² = 1.5625

Med: Large = 1.2² = 1.44

Small:Large = 1.5625 * 1.44 = 2.25

So if the Surface Area of the SMALL is 100 sq inches, then the Surface Area of the MEDIUM is

100 * 1.5625 = 156.25 square inches.

If the Surface Area of the MEDIUM is 156.25 sq inches, the the Surface Area of the LARGE is

156.25 * 1.44 = 225 square inches

Check: Small = 100 * 2.25 to find Large = 225 square inches.√

If we want the Volume scale factor, we must cube the Dimensions scale factor (b/c volume has CUBIC units)

Small:Med = 1.25³ = 1.953125

Med:Large = 1.2³ = 1.728

Small : Large = 1.953125 * 1.728 = 3.375

So if the Volume of the small is 1920 cubic inches, the the Volume of the Medium is

1920 * 1.953125 = 3750 cubic inches

If the Volume of the medium is 3750 cubic inches, then the Volume of the Large is

3750 * 1.728 = 6480 cubic inches

Check: Small 1920 * 3.375 (small to large scale factor) = 6480 cubic inches√

Now, lets talk about glass costs. The amount of glass needed for a fish tank is a SURFACE AREA problem. So we need to use the scale factors for the Surface Area to determine the costs of the glass in the Medium and Large tanks.

$12 for small * 1.5625 = $18.75

$18.75 for medium * 1.44 = $27.00

And I think that answers all of the questions.