In the xy-plane, the graph of the linear function f is perpendicular to the line with equa… — SAT

In the $xy$ -plane, the graph of the linear function $f$ is perpendicular to the line with equation $3x + 4y = 48$ . The graph of $f$ intersects the $x$ -axis at the point $(p, 0)$ and the $y$ -axis at the point $(0, q)$ , where $q > 0$ . If the distance between the points $(p, 0)$ and $(0, q)$ is $15$ , what is the value of $q$ ?

Cevap: 12

Cevap

First, find the slope of the given line by rewriting

3x + 4y = 48

in slope-intercept form:

y = -\frac{3}{4}x + 12

. The slope is

-\frac{3}{4}

. The graph of the linear function

f

is perpendicular to this line, so its slope is the negative reciprocal,

\frac{4}{3}

. With a

y

-intercept of

(0, q)

, the equation of

f

y = \frac{4}{3}x + q

. Setting

y = 0

gives the

x

-intercept

(p, 0) = (-\frac{3}{4}q, 0)

. The distance between these intercepts is

\sqrt{(-\frac{3}{4}q)^2 + q^2} = \sqrt{\frac{25}{16}q^2} = \frac{5}{4}q

(since

q > 0

). Given that the distance is

15

, we solve

\frac{5}{4}q = 15

to find

q = 12

Adım Adım Çözüm

Find the slope of the line

3x + 4y = 48

The slope of the line is

-\frac{3}{4}

To find the slope of the perpendicular line

f

, we first need the slope of the given line. Rewriting

3x + 4y = 48

in slope-intercept form gives

y = -\frac{3}{4}x + 12

Determine the slope of

f

The slope of

f

\frac{4}{3}

Perpendicular lines have slopes that are negative reciprocals of each other.

Find the

x

-intercept of

f

in terms of

q

The

x

-intercept of

f

(-\frac{3}{4}q, 0)

Since the

y

-intercept of

f

(0, q)

, the equation of

f

y = \frac{4}{3}x + q

. Setting

y = 0

gives

0 = \frac{4}{3}x + q

, which simplifies to

x = -\frac{3}{4}q

Use the distance formula between the intercepts to solve for

q

The value of

q

12

The distance between

(-\frac{3}{4}q, 0)

and

(0, q)

\sqrt{(-\frac{3}{4}q)^2 + q^2} = \sqrt{\frac{25}{16}q^2} = \frac{5}{4}q

since

q > 0

. Setting this distance to

15

gives

\frac{5}{4}q = 15

, which yields

q = 12

Anahtar Kavram

Properties of perpendicular lines, finding intercepts, and utilizing the distance formula in coordinate geometry.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru