A person stands $12$ meters east of an intersection and watches a car driving away from the intersection to the north at $4$ meters per second. $\newline$ At a certain instant, the car is $9$ meters from the intersection. $\newline$ What is the rate of change of the distance between the car and the person at that instant (in meters per second)? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{20}{3}$ $\newline$ (B) $15$ $\newline$ (C) $4 \sqrt{10}$ $\newline$ (D) $2$ . $4$

Full solution

Q. A person stands

12

meters east of an intersection and watches a car driving away from the intersection to the north at

4

meters per second.

\newline

At a certain instant, the car is

9

meters from the intersection.

\newline

What is the rate of change of the distance between the car and the person at that instant (in meters per second)?

\newline

Choose

1

answer:

\newline

(A)

\frac{20}{3}

\newline

(B)

15

\newline

(C)

4 \sqrt{10}

\newline

(D)

2

4

Find Hypotenuse Length: First, let's find the length of the hypotenuse using the Pythagorean theorem: $\text{hypotenuse}^2 = 9^2 + 12^2$ . That's $\text{hypotenuse}^2 = 81 + 144$ . So, $\text{hypotenuse}^2 = 225$ . Taking the square root, $\text{hypotenuse} = 15$ meters.
Calculate Rate of Change: Now, let's use the formula for the rate of change of the hypotenuse $\frac{dh}{dt}$ in a right triangle, which is given by $\frac{dh}{dt} = \frac{(\frac{dx}{dt} \cdot x + \frac{dy}{dt} \cdot y)}{h}$ , where $x$ and $y$ are the legs of the triangle and $\frac{dx}{dt}$ and $\frac{dy}{dt}$ are their rates of change. Here, $\frac{dx}{dt} = 0$ (because the person is not moving east or west), $\frac{dy}{dt} = 4$ m/s (the speed of the car going north), $x = 12$ meters, $y = 9$ meters, and $\frac{dh}{dt} = \frac{(\frac{dx}{dt} \cdot x + \frac{dy}{dt} \cdot y)}{h}$ $0$ meters.
Apply Formula: Plugging in the values, we get $\frac{dh}{dt} = \frac{(0 \times 12 + 4 \times 9)}{15}$ . That simplifies to $\frac{dh}{dt} = \frac{(0 + 36)}{15}$ . So, $\frac{dh}{dt} = \frac{36}{15}$ .
Simplify Result: Now, let's simplify $36 / 15$ . That's $\frac{dh}{dt} = 2.4$ meters per second.