Question

Geometry question, any help is appreciated

Answer 1

Volume ratio 250 : 686 = 125 : 343
Radius ratio ∛(125) : ∛(343) = 5 : 7 scale factor
Area ratio (5)² : (7)² = 25 : 49

Scale factor (5:7), area ratio (25:49), volume ratio (125:343)

Answer 2

Hence, the ratios are given by 5:7, 25:49, 125:343.

Explanation:

We are given two spheres such that:

Volume of small sphere
`V_(s)` = 250
`yd^(3)`

Volume of large sphere
`V_(l)` = 686
`yd^(3)`.

Then the ratio of volume = 250 : 686 = 125 : 343.

Since, the volume of a sphere =
`(4 \pi * r^(3))/(3)`, this gives us that the ratio of the radius = ∛
`V_(s)` : ∛
`V_(l)`

i.e. The ratio of the radius = ∛125 : ∛343 = 5 : 7.

Further, as the surface area of a sphere =
`4 \pi * r^(2)`, this gives us that the ratioof surface area =
`r_(s) ^(2)` :
`r_(l) ^(2)`

i.e The ratio of the surface area =
`5^(2)` :
`7^(2)` = 25 : 49

So, the ratio of the surface areas = 25 : 49.

Hence, the ratios are given by 5:7, 25:49, 125:343.