A $10$ -meter ladder is sliding down a vertical wall so the distance between the top of the ladder and the ground is decreasing at $3$ meters per minute. $\newline$ At a certain instant, the bottom of the ladder is $6$ meters from the wall. $\newline$ What is the rate of change of the area formed by the ladder at that instant (in square meters per minute)? $\newline$ Choose $1$ answer: $\newline$ (A) $-\frac{7}{2}$ $\newline$ (B) $7$ $\newline$ (C) $12$ $\newline$ (D) $-6$

Full solution

Q. A

10

-meter ladder is sliding down a vertical wall so the distance between the top of the ladder and the ground is decreasing at

3

meters per minute.

\newline

At a certain instant, the bottom of the ladder is

6

meters from the wall.

\newline

What is the rate of change of the area formed by the ladder at that instant (in square meters per minute)?

\newline

Choose

1

answer:

\newline

(A)

-\frac{7}{2}

\newline

(B)

7

\newline

(C)

12

\newline

(D)

-6

Denote Variables: Let's denote the distance from the top of the ladder to the ground as $y$ (in meters). The ladder's length is $10$ meters. We're given that $y$ is decreasing at $3$ meters per minute, so $\frac{dy}{dt} = -3$ m/min.
Calculate Triangle Area: The area of the right-angled triangle formed by the wall, ground, and ladder can be expressed as $A = \frac{1}{2} \times \text{base} \times \text{height}$ . Here, the base is the distance from the wall, which is $6$ meters, and the height is $y$ . So, $A = \frac{1}{2} \times 6 \times y$ .
Find Rate of Change: To find the rate of change of the area, we need to differentiate $A$ with respect to time ( $t$ ). So, $\frac{dA}{dt} = (\frac{1}{2}) \times 6 \times \frac{dy}{dt}$ . We substitute $\frac{dy}{dt}$ with $-3$ to get $\frac{dA}{dt} = (\frac{1}{2}) \times 6 \times (-3)$ .
Check Right-Angled Triangle: Calculating $\frac{dA}{dt}$ gives us $\frac{dA}{dt} = \frac{1}{2} \times 6 \times (-3) = -9$ square meters per minute. But wait, we didn't use the ladder's length to ensure the triangle is right-angled. We need to check if the ladder's length and the distances given form a right-angled triangle.