make a table that compares the surface areas and the volumes of the seven dolls

Question

There are seven dolls. the largest doll is 7 inches. each of the other dolls is 1 inch shorter than the next larger doll. make a table that compares the surface areas and the volumes of the seven dolls.

Answer 1

This is an exercise in ratio reasoning. You don't need to calculate anything, you need to give the ratios that the areas and volumes change by.  The reasoning goes like this:

Height is a linear function.

Area is a square function; so if the height changes by some ratio, the Area changes by that ratio squared.

Volume is a cube function; so if the height changes by some ratio, the Volume changes by that ratio cubed.

You could also go into if the Area changed how much does the volume change, but that is a little harder and not needed here.

 

SO if the height of something doubles (x 2) the surface goes up by (x 2)² or (x 4).

If the height of something triples (x 3) the surface goes up by (x 3)² or (x 9).

If the height of something halves (x 1/2) the surface goes down by (x 1/2)² or (x 1/4).

The same thing goes for volumes except all the squares become cubes.

If the height of something doubles (x 2) the volume goes up by (x 2)³ or (x 8.)

 

For you your ratios are not all whole numbers like I showed.

If the 7" doll has  an area = A  and a volume = V

    then if the height was 1/2 the area would be A/2 and the volume V/8

But you been to work with ratios like 4/7, 2/7, ...

 

H   factor         Area        Volume

7       1              A                V

6      6/7        (6/7)²A       (6/7)³V

5      5/7        (5/7)²A       (5/7)³V

4

 

The factors are 6/7 & 5/7 are because the dolls are 6/7 & 5/7 of the original height.

Maybe it is okay to leave it like that, or maybe you should calculate the numbers.

Like (6/7)² = some decimal number.  (What are your directions?)

 

Try to do the rest of the table.

If you get stuck come back and let us know what you are stuck on.