the surface areas of two similar solids are 340 yd2 and 1158 yd2 the volume of the larger solids is 1712 yd3 what is the volume of the smaller solid?
the surface area of two similar objects is proportioal to the square of their sides
the volume of two similar objects is proportional to the cube of their sides
so ratio of sides of smaller : larger = √340 : √1158
so volume smaller/volume of larger = (√340)^3 / (√1158)^3
volume of smaller = 1712(√340)^3 / (√1158)^3
= appr 272.37 yds^3
To find the volume of the smaller solid, we can use the concept of similarity between the two solids.
According to the concept of similarity, the ratio of the surface areas of two similar solids is equal to the square of the ratio of their corresponding side lengths.
Let's denote the surface areas of the smaller and larger solids as S1 and S2, respectively. Also, let's denote the volumes of the smaller and larger solids as V1 and V2, respectively.
The given surface areas are:
S1 = 340 yd²,
S2 = 1158 yd².
The given volume of the larger solid is:
V2 = 1712 yd³.
We can set up the following equation using the surface area ratios:
(S1/S2)² = (V1/V2).
Substituting the given values, we have:
(340/1158)² = V1/1712.
Simplifying the equation, we find:
(0.2938)² = V1/1712.
0.08618 = V1/1712.
Multiplying both sides by 1712, we get:
V1 = 1712 * 0.08618.
V1 ≈ 147.8 yd³.
Therefore, the volume of the smaller solid is approximately 147.8 yd³.
To find the volume of the smaller solid, we need to set up a proportion using the surface areas of the two similar solids.
The surface areas are given as 340 yd² for the smaller solid and 1158 yd² for the larger solid.
First, let's set up the proportion:
(surface area of smaller solid) / (surface area of larger solid) = (volume of smaller solid) / (volume of larger solid)
340 / 1158 = (volume of smaller solid) / 1712
Now we can solve for the volume of the smaller solid:
(volume of smaller solid) = (340 / 1158) * 1712
Calculating this, we get:
(volume of smaller solid) ≈ 500 yd³
Therefore, the volume of the smaller solid is approximately 500 yd³.