The volumes of two similar solids are 857.5mm^3 and 540 mm^3. The surface area of the smaller solid is 108 mm^2 what is the surface area of the larger solid
147mm^2
I apologize for the error in my previous response. The correct calculation is:
Surface area of larger solid = (1.172)^2 * 108
Surface area of larger solid ≈ 147 mm^2
Therefore, the surface area of the larger solid is approximately 147 mm^2.
Let two corresponding sides of the shapes be x and y , x > y
from volume data:
x^3 / y^3 = 857.5mm^3 / 540 mm^3
(x/y)^3 = 42.875 / 27
x/y = 1.1666... = 1 1/6 = 7/6
SA larger / SA smaller = 7^2/6^2 = 49/36
SA larger/108 = 49/36
SA larger = (49/36)(108) = exactly 147 , as Pinky had stated
Thank you for correcting the calculation and providing the complete solution. That is correct, the surface area of the larger solid is exactly 147 mm^2.
To find the surface area of the larger solid, we need to use the concept of similarity between the two solids.
First, let's establish the relationship between the volumes of similar solids. We know that the ratio of the volumes of two similar solids is equal to the cube of the ratio of their corresponding side lengths. In other words:
(Volume of larger solid) / (Volume of smaller solid) = (Side length of larger solid / Side length of smaller solid)^3
Now, let's take the ratios of the given volumes:
(Volume of larger solid) / (Volume of smaller solid) = 857.5 mm^3 / 540 mm^3
Simplifying this, we get:
(Volume of larger solid) / (Volume of smaller solid) = 1.586...
Next, we can solve for the ratio of the side lengths by taking the cube root of the volume ratio:
(Side length of larger solid / Side length of smaller solid) ≈ ∛(1.586) ≈ 1.137
So, the side length of the larger solid is approximately 1.137 times longer than the side length of the smaller solid.
To find the surface area ratio between similar solids, we square the ratio of their side lengths. Therefore:
(Surface area of larger solid) / (Surface area of smaller solid) = (Side length of larger solid / Side length of smaller solid)^2
Plugging in the values we found earlier:
(Surface area of larger solid) / (108 mm^2) = 1.137^2
Simplifying this, we get:
(Surface area of larger solid) ≈ 108 mm^2 × 1.292
Calculating that, we find:
(Surface area of larger solid) ≈ 139.3 mm^2
Therefore, the surface area of the larger solid is approximately 139.3 mm^2.
We know that the volumes of two similar solids are proportional to the cube of the ratio of their corresponding lengths. Let's assume that the ratio of their corresponding lengths is "x". Then:
(x)^3 = 857.5 / 540
x = (857.5 / 540)^(1/3)
x ≈ 1.172
This means that the larger solid is 1.172 times larger than the smaller solid. Therefore, if the surface area of the smaller solid is 108 mm^2, the surface area of the larger solid would be:
Surface area of larger solid = (1.172)^2 * 108
Surface area of larger solid ≈ 144 mm^2
Thus, the surface area of the larger solid is approximately 144 mm^2.