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stepan [7]
3 years ago
15

What is the surface area

Mathematics
1 answer:
anygoal [31]3 years ago
3 0
Bear in mind that the area of a "parallelogram" is base * height.

notice the picture, the front and back parallelograms, have an altitude/height of 3 cm, and  a base of 8 cm, so the area of one of those two is 3*8.

as far as the left and right parallelograms, they have a base of 7 cm, but the altitude is also 3 cm, so the area of one of those is 3 *7.

bear in mind the altitude/height is the distance from the base to the top, and in this case is just 3 cm.

anyway, you have 4 parallelograms, get the area of each, add them up, and that's the surface area of the figure.

2(3*8) + 2(3*7).
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Consider the nth partial sum, denoted by S_n:

S_n=\dfrac1{16}\left(\dfrac34\right)+\dfrac1{16}\left(\dfrac34\right)^2+\dfrac1{16}\left(\dfrac34\right)^3+\cdots+\dfrac1{16}\left(\dfrac34\right)^n

Multiply both sides by \frac34:

\dfrac34S_n=\dfrac1{16}\left(\dfrac34\right)^2+\dfrac1{16}\left(\dfrac34\right)^3+\dfrac1{16}\left(\dfrac34\right)^4+\cdots+\dfrac1{16}\left(\dfrac34\right)^{n+1}

Subtract S_n from this:

\dfrac34S_n-S_n=\dfrac1{16}\left(\dfrac34\right)^{n+1}-\dfrac1{16}\left(\dfrac34\right)

Solve for S_n:

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\displaystyle\lim_{n\to\infty}S_n=\lim_{n\to\infty}\sum_{k=1}^n\frac{3^k}{4^{k+2}}=\sum_{k=1}^\infty\frac{3^k}{4^{k+2}}=\frac3{16}

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