QuestionHow many times greater is the surface area of Seung's square pyramid block than Derek's half-sized block? (A) 2 (B) 3 (C) 4 (D) 8

Studdy Solution

STEP 1

Assumptions1. The shape of the block is a square pyramid. . The dimensions of Derek's block are half of Seung's block.
3. We are comparing the surface areas of the two blocks.

STEP 2

First, let's understand the formula for the surface area of a square pyramid. The surface area is the sum of the area of the base (which is a square) and the area of the four triangular faces.
$Surface\, Area = Base\, Area +4 \times (Area\, of\, one\, triangular\, face)$

STEP 3

The area of a square is side length squared, and the area of a triangle is half the base times the height. So, for a square pyramid, the surface area formula becomes $Surface\, Area = s^2 + \times (0.5 \times s \times sl)$ where s is the side length of the base and sl is the slant height of the pyramid.

STEP 4

Now, let's denote the side length of Seung's block as and the side length of Derek's block as D. Since Derek's block has dimensions that are half of Seung's block, we have D =/2.

STEP 5

Let's calculate the surface area of Seung's block using the formula. We'll denote this as SA.
$SA =^2 +4 \times (0.5 \times \times)$

STEP 6

Now, let's calculate the surface area of Derek's block using the formula. We'll denote this as SA_D.
$SA_D = D^2 +4 \times (0.5 \times D \times D)$

STEP 7

But we know that D =/2, so we can substitute this into the equation for SA_D.
$SA_D = (/2)^2 +4 \times (0.5 \times (/2) \times (/2))$

STEP 8

implify the equation for SA_D.
$SA_D =^2/4 +^2/2$

STEP 9

Now, we want to find how many times greater the surface area of Seung's block is than Derek's block. We can do this by dividing SA by SA_D.
$Ratio = SA / SA_D$

STEP 10

Substitute the equations for SA and SA_D into the equation for the ratio.
$Ratio = (^2 +2^2) / (^2/4 +^2/2)$

STEP 11

implify the equation for the ratio.
$Ratio =4$ So, the surface area of Seung's block is4 times greater than the surface area of Derek's block. Therefore, the answer is (C)4.

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