1967 AMC 12 Problem 12

Attempt Problem 12 of the 1967 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 1967 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

12.

If the (convex) area bounded by the

x

-axis and the lines

y=mx+4,

x=1,

and

x=4

7,

then

m

equals:

-\dfrac12

-\dfrac23

-\dfrac32

-2

none of these

The parallel vertical sides have lengths

m+4

and

4m+4

Their separation is

3

, so use the trapezoid-area formula

The bounded region is a trapezoid with parallel sides

m+4

and

4m+4,

separated by

3.

Thus

\frac32\bigl((m+4)+(4m+4)\bigr)=7.

Hence

15m+24=14,

m=-\frac{2}{3}.

The endpoint heights are positive, as required for the stated convex region.

Therefore, the correct answer is B.

Problem 12 in other years