The surface area of two similar solids are 384 yd² and 1,057 yd². The volume of the larger solid is 1,796 yd³. What is the volume of the smaller solid?
a. 1,795 yd³
b. 1,082 yd³
c. 393 yd³
d. 978 yd³
We know that the ratio of surface areas is the square of the ratio of lengths of corresponding sides, and the ratio of volumes is the cube of the ratio of lengths of corresponding sides. Let x be the ratio of lengths of corresponding sides (i.e. the scale factor) between the smaller and larger solids.
Then we have:
(x²) * 384 = 1057
x = √(1057/384) ≈ 1.446
And:
(x³) * V = 1796
V = 1796 / (x³) = 1796 / (1.446³) ≈ 1082
So the volume of the smaller solid is approximately 1082 yd³, which is choice (b).
To find the volume of the smaller solid, we need to use the concept of ratio and proportion.
We are given that the surface areas of the two similar solids are 384 yd² and 1,057 yd². Let's call the surface area of the smaller solid A₁ and the surface area of the larger solid A₂.
We know that the ratio of the surface areas is equal to the square of the ratio of the linear dimensions (sides).
(A₁ / A₂) = (s₁ / s₂)²
Let's denote the linear dimensions of the smaller solid as s₁, and the linear dimensions of the larger solid as s₂.
Now, we can set up a proportion using the given surface areas:
384 / 1,057 = (s₁ / s₂)²
To find the linear dimensions of the larger solid, we need to take the square root of the ratio of the surface areas:
√(384 / 1,057) = s₁ / s₂
s₁ / s₂ = 0.6201 (rounded to 4 decimal places)
Now, we have the ratio of the linear dimensions.
To find the volume ratio, we cube the ratio of the linear dimensions:
(s₁ / s₂)³ = (0.6201)³ = 0.2384
The volume of the larger solid is 1,796 yd³. Let's call the volume of the smaller solid V₁ and the volume of the larger solid V₂.
V₁ = (V₂ / (s₁ / s₂)³) = (1,796 / 0.2384) = 7,535.0337 yd³ (rounded to 4 decimal places)
Therefore, the volume of the smaller solid is approximately 7,535.0337 yd³ which is closest to option:
b. 1,082 yd³