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A 5-meter ladder is sliding down a vertical wall so the distance between the bottom of the ladder and the wall is increasing at 6 meters per minute.
At a certain instant, the top of the ladder is 3 meters from the ground.
What is the rate of change of the area formed by the ladder at that instant (in square meters per minute)?
Choose 1 answer:
(A) 6
(B) -14
(C)
18
(D) -7

A $5$ -meter ladder is sliding down a vertical wall so the distance between the bottom of the ladder and the wall is increasing at $6$ meters per minute. $\newline$ At a certain instant, the top of the ladder is $3$ meters from the ground. $\newline$ 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) $6$ $\newline$ (B) $-14$ $\newline$ (C) $\mathbf{1 8}$ $\newline$ (D) $-7$

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Calculate unit rates with fractions

Full solution

Q. A

5

-meter ladder is sliding down a vertical wall so the distance between the bottom of the ladder and the wall is increasing at

6

meters per minute.

\newline

At a certain instant, the top of the ladder is

3

meters from the ground.

\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)

6

\newline

(B)

-14

\newline

(C)

\mathbf{1 8}

\newline

(D)

-7

Form Right Triangle: The ladder, wall, and ground form a right triangle. The ladder is the hypotenuse, and the distance from the wall is one leg, which is increasing at $6$ meters per minute. The other leg is the height from the ground to the top of the ladder, which is $3$ meters.
Calculate Area: Let's call the distance from the wall to the bottom of the ladder $x$ meters. The area $A$ of the triangle formed by the ladder, wall, and ground is $(\frac{1}{2}) \times x \times 3$ .
Differentiate Area: Differentiate the area with respect to time to find the rate of change of the area, $\frac{dA}{dt}$ . $\frac{dA}{dt} = \frac{1}{2} \times 3 \times \frac{dx}{dt}$ .
Find Rate of Change: We know $\frac{dx}{dt}$ , the rate at which the bottom of the ladder moves away from the wall, is $6$ meters per minute. So, $\frac{dA}{dt} = \frac{1}{2} \times 3 \times 6$ .
Calculate $\frac{dA}{dt}$ : Calculate $\frac{dA}{dt}$ : $\frac{dA}{dt} = \left(\frac{1}{2}\right) \times 3 \times 6 = 3 \times 6 = 18$ square meters per minute.

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