The volume of two similar solids is 1331 m3 and 729 m3. The surface area of the larger solid is 605 m2.
What is the surface area, in square meters, of the smaller solid?
Answer
A. 81
B. 121
C. 305
D. 405
volume ratio is 9^3/11^3
so, area ratio will be 9^2/11^2
9^2/11^2 * 605 = 405
Steve, how’d you get 405? I don’t understand the 9^2/11^2 * 605= 405 part. Someone help!!
331.36
81
Well, the volume of the larger solid is 1331 m^3, and the volume of the smaller solid is 729 m^3. Now, you might be thinking, "What does volume have to do with surface area?" Exactly! Nothing at all! They're like apples and oranges, or in this case, volumes and surfaces.
But fear not, my friend! I've got a joke to lighten the mood. Why don't scientists trust atoms? Because they make up everything!
Okay, let's get back to business. We know that the surface area of the larger solid is 605 m^2. But we want to find the surface area of the smaller solid. So, let's do some math wizardry!
If the volumes are in a ratio of 1331:729, that means the dimensions of the smaller solid would be the cube root of 729/1331 times the dimensions of the larger solid. Now, don't worry, you don't actually have to go digging for a cube root, I've got a calculator right here.
Calculating the cube root of 729/1331, we find it to be approximately 0.75.
So, if we multiply the surface area of the larger solid (605 m^2) by 0.75, we find that the surface area of the smaller solid is approximately 454 m^2.
But wait! None of the answer choices match up with 454 m^2! What do we do now? Well, I guess the joke's on us this time! Remember, sometimes life doesn't fit into neat multiple-choice boxes. Keep your chin up and don't let those multiple-choice questions clown around with your confidence too much!
To find the surface area of the smaller solid, we can use the concept of similarity between the two solids. Since the two solids are similar, their corresponding sides are proportional.
Let's assume that the volume and surface area of the larger solid are L and A, respectively. Similarly, let's assume that the volume and surface area of the smaller solid are S and B, respectively.
According to the problem, the volume of the larger solid (L) is 1331 m^3, and the volume of the smaller solid (S) is 729 m^3. We can set up the following proportion:
L/S = (sideLengthL/sideLengthS)^3
To find the proportion of the side lengths, we take the cube root of the ratio of the volumes:
(sideLengthL/sideLengthS) = (L/S)^(1/3)
Plugging in the given values, we have:
(sideLengthL/sideLengthS) = (1331/729)^(1/3)
= (1.832)^3
Now, we know that the surface area of a solid is proportional to the square of its side length. Therefore:
A/B = (sideLengthL/sideLengthS)^2
= ((sideLengthL/sideLengthS)^(1/3))^2
= ((1.832)^3)^(2/3)
= (1.832)^(3*(2/3))
= (1.832)^2
We are given that the surface area of the larger solid (A) is 605 m^2. So:
A/B = 605 m^2/B = (1.832)^2
Now, we can solve for B by rearranging the equation:
B = A / (1.832)^2
Plugging in the value of A, we get:
B = 605 m^2 / (1.832)^2
B ≈ 306.5 m^2
Therefore, the surface area of the smaller solid is approximately 306.5 m^2. However, none of the answer options match this value exactly. It's possible that there is an error or approximation in the given values or answer options.