2022 AMC 12B Problem 14

Attempt Problem 14 of the 2022 AMC 12B 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 2022 AMC 12B solutions, or check the answer key.

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

14.

The graph of

y = x^2 + 2x - 15

intersects the

x

-axis at points

A

and

C

and the

y

-axis at point

B.

What is

\tan(\angle ABC)?

\dfrac17

\dfrac14

\dfrac37

\dfrac12

\dfrac47

Factor to find the intercepts:

A

and

C

on the

x

-axis,

B

on the

y

-axis

With vectors

\vec{BA}

and

\vec{BC},

use

\tan\theta = \dfrac{|\vec{BA} \times \vec{BC}|}{\vec{BA} \cdot \vec{BC}}

Factoring,

x^2 + 2x - 15 = (x+5)(x-3),

A = (-5, 0)

and

C = (3, 0),

and the

y

-intercept is

B = (0, -15).

Then

\vec{BA} = (-5, 15)

and

\vec{BC} = (3, 15).

Using the cross and dot products,

\begin{gathered} \tan(\angle ABC) = \scriptsize \dfrac{|(-5)(15) - (15)(3)|}{(-5)(3) + (15)(15)} \\ = \dfrac{120}{210} \\ = \dfrac47. \end{gathered}

Thus, the correct answer is E.

Problem 14 in other years