1969 AMC 12 Problem 22

Attempt Problem 22 of the 1969 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1969 AMC 12 solutions, or check the answer key.

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22.

Let

K

be the measure of the area bounded by the

x

-axis, the line

x=8,

and the curve defined by

\begin{aligned} f=\{(x,y)\mid{}&\\ y&=x\\ &\text{when }0\le x\le5,\\ y&=2x-5\\ &\text{when }5\le x\le8\}. \end{aligned}

Then

K

is:

21.5

36.4

36.5

44

less than

44

but arbitrarily close to it

Split the region at

x=5

Use a triangle from

0

5

and a trapezoid from

5

8

From

x=0

5,

the region is a triangle of area

\frac12(5)(5)=12.5.

From

x=5

8,

the endpoint heights are

5

and

11,

so the trapezoid has area

\frac12(5+11)(3)=24.

Thus

K=12.5+24=36.5.

Therefore, the correct answer is C.

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