The values of two similar solids are 857.5 mm cubed and 540 mm cubed the surface area of the smaller solid is 108 mm squared what is the surface area of the larger solid
A 147 mm2
B 68mm2
C. 16mm2
D. 216 mm2
We can use the fact that the ratios of corresponding lengths, areas, and volumes of similar solids are equal. Let's denote the surface area of the larger solid with $A$. We have:
$$\frac{\text{surface area of smaller solid}}{\text{surface area of larger solid}}=\frac{108}{A}$$
Since the solids are similar, their volumes are in the same ratio as the cubes of their corresponding side lengths. Let's denote the side lengths of the smaller solid with $x$, then the side length of the larger solid is $\frac{x}{\sqrt[3]{\frac{857.5}{540}}}$. We have:
$$\frac{\text{volume of smaller solid}}{\text{volume of larger solid}}=\left(\frac{x}{\frac{x}{\sqrt[3]{\frac{857.5}{540}}}}\right)^3=\frac{540}{857.5}$$
Simplifying, we get:
$$\frac{\sqrt[3]{\frac{857.5}{540}}^3}{1}=\frac{\sqrt[3]{857.5}}{\sqrt[3]{540}}=\frac{\sqrt[3]{857.5/540}}{1}$$
Multiplying the two ratios, we obtain:
$$\frac{\text{surface area of smaller solid}}{\text{surface area of larger solid}}\cdot\frac{\text{volume of smaller solid}}{\text{volume of larger solid}}=\frac{108}{A}\cdot\frac{540}{857.5/540}=\frac{108\cdot 540\cdot 540}{857.5}\approx 513.6$$
Setting this equal to the ratio of the surface areas of the similar solids, we get:
$$\frac{A}{108}=\frac{857.5}{540}\cdot\frac{1}{\frac{108\cdot 540\cdot 540}{857.5}}\approx 0.45195$$
Therefore, $A\approx 0.45195\cdot 108\approx\boxed{48.84\text{ mm}^2}$, which is closest to answer choice $\textbf{(A)}$.
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To solve this problem, we need to understand the concept of similar solids.
Similar solids are objects that have the same shape but not necessarily the same size. When two solids are similar, their corresponding sides are proportional. This means that if one side of the smaller solid is x times smaller than the corresponding side of the larger solid, then all other corresponding sides will also be x times smaller.
Let's assume that the corresponding sides of the smaller and larger solids are in the ratio of x:1, where x is the proportionality constant.
Given that the volume of the smaller solid is 857.5 mm³ and the volume of the larger solid is 540 mm³, we can set up the following equation using the volume formula for similar solids:
(x^3)(857.5) = (1^3)(540)
Simplifying the equation, we get:
x^3 = 0.63
Taking the cube root of both sides, we find:
x ≈ 0.892
So, the corresponding sides of the smaller solid are approximately 0.892 times smaller than the corresponding sides of the larger solid.
Now, let's find the ratio of the surface areas of the smaller and larger solids. Since the ratio of corresponding sides is x:1, the ratio of the surface areas will be (x^2):(1^2), which simplifies to x^2.
Given that the surface area of the smaller solid is 108 mm², we can set up the following equation:
(x^2)(108) = (1^2)(A),
Where A represents the surface area of the larger solid that we need to find.
Substituting the value of x, we have:
(0.892^2)(108) = A
Simplifying and calculating, we find:
A ≈ 68 mm²
Therefore, the surface area of the larger solid is approximately 68 mm².
Hence, the correct answer is B. 68mm².