Math  /  Geometry

QuestionThe two solids are similar, and the ratio between the lengths of their edges is 2:72: 7. What is the ratio of their surface areas? A. 2:72: 7 B. 4:144: 14 C. 8:3438: 343 D. 4:494: 49

Studdy Solution

STEP 1

What is this asking? If we know how the lengths of the edges of two similar solids compare, how do their surface areas compare? Watch out! The ratio of surface areas isn't the same as the ratio of lengths!

STEP 2

1. Relate the ratio of lengths to the ratio of areas.

STEP 3

Let's call the ratio of the lengths of corresponding edges kk.
In our case, we're given that k=27k = \frac{2}{7}.
This means if we pick an edge on the smaller solid with length l1l_1, the corresponding edge on the larger solid will have length l2=72l1l_2 = \frac{7}{2} \cdot l_1.
So, kk tells us how much bigger or smaller one solid is compared to the other!

STEP 4

Now, think about the surface area.
The surface area of any solid is just the sum of the areas of its faces.
Each face is some kind of 2D shape, like a square, triangle, or something more complicated.
The important thing is that the area of each face will be proportional to the *square* of its lengths.
For example, if you have a square with side length ss, its area is s2s^2.
If you double the side length to 2s2s, the area becomes (2s)2=4s2(2s)^2 = 4s^2, which is 4 times bigger!

STEP 5

So, if the ratio of corresponding lengths is kk, then the ratio of corresponding areas (like the areas of the faces) will be k2k^2.
This is because areas depend on lengths squared!

STEP 6

Since the ratio of the lengths is k=27k = \frac{2}{7}, the ratio of the surface areas will be k2=(27)2=2272=449k^2 = \left(\frac{2}{7}\right)^2 = \frac{2^2}{7^2} = \frac{4}{49}.

STEP 7

The ratio of the surface areas of the two similar solids is **4:49**, so the answer is **D**.

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