Question

16
a ladder is leaning against a wall. paul has measured and determined that the distance along the ground from the base of the ladder to the wall is eight inches less than the distance along the wall from the ground to the top of the ladder.
which of the following functions describes the length of the ladder in terms of x, the distance along the wall from the ground to the top of the ladder?
a. $l(x)=\sqrt{2x - 8}$
b. $l(x)=\sqrt{2x^2 + 16x + 64}$
c. $l(x)=\sqrt{2x^2 - 64}$
d. $l(x)=\sqrt{2x^2 - 16x + 64}$

Explanation:

Step1: Define variables

Let $x$ = distance from ground to top of ladder on wall (in inches). Ground distance = $x - 8$ inches.

Step2: Apply Pythagorean theorem

Ladder length $L(x)$ is hypotenuse: $L(x) = \sqrt{x^2 + (x-8)^2}$

Step3: Expand the squared term

$(x-8)^2 = x^2 - 16x + 64$

Step4: Simplify the radicand

$x^2 + x^2 - 16x + 64 = 2x^2 - 16x + 64$

Step5: Write final function

$L(x) = \sqrt{2x^2 - 16x + 64}$

Answer:

D. $L(x) = \sqrt{2x^2 - 16x + 64}$