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 yd²
b. 1,082 yd²
c. 393 yd²
d. 978 yd²
Since the solids are similar, the ratio of their surface areas is equal to the square of the ratio of their lengths. Let x be the ratio of the lengths. Then we have:
(x²)384 = 1,057
Solving for x, we get:
x = √(1,057/384) ≈ 1.5
So the lengths of the two solids are in a 1:1.5 ratio. Since the volume is proportional to the cube of the length, the ratio of the volumes is:
(1.5³) = 3.375
So the volume of the smaller solid is:
1,796/3.375 ≈ 532 yd³
Therefore, the answer is not among the choices given.
To find the volume of the smaller solid, we can use the concept of similar solids. The surface area is directly proportional to the square of the lengths, and the volume is directly proportional to the cube of the lengths.
We can set up a proportion using the surface areas:
384/1057 = (side length of smaller solid)² / (side length of larger solid)²
To find the side length of the smaller solid, we can take the square root of both sides:
(sqrt 384/1057) = (side length of smaller solid) / (side length of larger solid)
Now, we can find the side length of the smaller solid:
(sqrt 384/1057) = (side length of smaller solid) / (side length of larger solid)
(sqrt 384/1057) * (side length of larger solid) = side length of smaller solid
Once we have the side lengths, we can find the volume by using the fact that the volume of the larger solid is 1,796 yd³:
(volume of larger solid) / (volume of smaller solid) = (side length of larger solid)³ / (side length of smaller solid)³
1796 / (volume of smaller solid) = (side length of larger solid)³ / (side length of smaller solid)³
Now we can rearrange the equation to solve for the volume of the smaller solid:
(volume of smaller solid) = 1796 * (side length of smaller solid)³ / (side length of larger solid)³
Finally, we can substitute the side lengths we found earlier into the equation to calculate the volume of the smaller solid.