The surface areas of two similar solids are 416yd ^2 and 1,354yd ^2. The volume of the larger solid is 2,976 yd^3. What is the volume of the smaller solid?

2,976 yd ^3
2,619 yd ^3
914 yd ^3
507 yd ^2

The areas of similar objects are proportional to the square of their sides

so 416/1354 = x^2/y^2 , where x and y are corresponding sides , x as the smaller
x^2/y^2 = 416/1354 = 208/677
x/y = √208/√677

the volumes of similar objects are proportional to the cubes of their corresponding sides
x^3/y^3 = volume/2976
(√208)^3 / (√677)^3 = volume/2976
volume = appr 506.8 yds^3

Your answer has yds^2, clearly a typo

Well, since the two solids are similar, their surface areas have a ratio equal to the square of the ratio of their volumes. We can set up a proportion to find the ratio:

416yd^2 / 1354yd^2 = V1^2 / (2976yd^3)^2

Now let's solve for V1:

V1^2 = 416yd^2 / 1354yd^2 * (2976yd^3)^2

Hmm, I'm scratching my circuits here, but this proportion doesn't seem to have a volume you provided. It seems that the smaller solid's volume is missing from the answer choices. I apologize for the confusion!

To find the volume of the smaller solid, we can set up a proportion using the surface areas and volumes of the two solids.

The ratio of the surface areas is 416yd^2 / 1,354yd^2. Simplifying this ratio, we get 0.307.
The ratio of the volumes is equal to the ratio of the cube of the corresponding sides in the similar solids.
Let x be the length of a side of the smaller solid. The length of a side of the larger solid would be (x * (2,976yd^3)^(1/3)).
So the ratio of the volumes is (x / (x * (2,976yd^3)^(1/3)) = 1 / (2,976yd^3)^(1/3).
Setting up the proportion, we have:
0.307 = 1 / (2,976yd^3)^(1/3)
To find the volume of the smaller solid, we need to solve for x.
Multiply both sides of the equation by (2,976yd^3)^(1/3):
0.307 * (2,976yd^3)^(1/3) = 1
Divide both sides of the equation by 0.307:
(2,976yd^3)^(1/3) = 1 / 0.307
Cube both sides of the equation to eliminate the cube root:
2,976yd^3 = (1 / 0.307)^3
Evaluate the right side of the equation:
2,976yd^3 = 8,454.35
Finally, divide both sides of the equation by 2,976 to solve for the volume of the smaller solid:
Volume of the smaller solid = 8,454.35 / 2,976 ≈ 2,827.46 yd^3

Therefore, the volume of the smaller solid is approximately 2,827.46 yd^3.

To solve this problem, we need to use the concept of similarity between two solids. When two solids are similar, their corresponding sides are proportional. In this case, we have the surface areas of two similar solids, which means the corresponding sides of the solids are also proportional.

Let's assume the surface area of the larger solid is A1, the surface area of the smaller solid is A2, the volume of the larger solid is V1, and the volume of the smaller solid is V2. We are given A1 = 1,354 yd^2, A2 = 416 yd^2, and V1 = 2,976 yd^3.

The ratio of the surface areas of two similar solids is equal to the square of the ratio of their corresponding side lengths. Mathematically, it can be written as:

(A1/A2) = (l1^2/l2^2)

Where l1 and l2 are the length of the corresponding sides of the solids.

Let's find the ratio of the surface areas:

(A1/A2) = (1,354/416)

Now, let's solve for the ratio of the length:

(1,354/416) = (l1^2/l2^2)

Cross-multiplying, we get:

1,354 * l2^2 = 416 * l1^2

Dividing both sides by l2^2, we have:

l1^2 = (1,354 * l2^2) / 416

Now, let's solve for the ratio of the volumes:

(V1/V2) = (l1^3/l2^3)

Substituting the value of l1^2 from the previous equation, we have:

(V1/V2) = ([(1,354 * l2^2) / 416]^3/l2^3)

Simplifying, we obtain:

(V1/V2) = [(1,354^3 * l2^6) / (416^3 * l2^3)]

Cross-multiplying, we get:

V2 * (1,354^3 * l2^6) = V1 * (416^3 * l2^3)

Dividing both sides by (1,354^3 * l2^6), we have:

V2 = (V1 * (416^3 * l2^3)) / (1,354^3 * l2^6)

Canceling out the l2^3 term, we get:

V2 = (V1 * 416^3) / 1,354^3

Substituting the given values, we have:

V2 = (2,976 * 416^3) / 1,354^3

Evaluating this expression gives:

V2 ≈ 2,619 yd^3

Therefore, the volume of the smaller solid is approximately 2,619 yd^3.