Question

If a balloon flying at an altitude of 30 m from an observer at a particular instant is moving horizontally at the rate of 1 m/s away from him, then the rate at which the balloon is moving away directly from the observer at the 40th second is (in m/s)

• A. 1.2
• B. 0.9
• C. 0.6
• D. 0.8

Hint:

In related rates problems, the general procedure is: 1. Identify the given rates and the rate you need to find. 2. Write an equation that relates the variables. 3. Differentiate the equation implicitly with respect to time. 4. Substitute the known values for the variables and their rates at the specific instant to solve for the unknown rate.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
This is a related rates problem. We are given the rate of change of one quantity (horizontal distance) and need to find the rate of change of another related quantity (direct distance). The relationship between the quantities can be modeled using a right-angled triangle.
Step 2: Key Formula or Approach
1. Draw a diagram representing the observer, the ground, and the balloon. This forms a right-angled triangle. 2. Let $h$ be the constant altitude of the balloon ($h=30$ m). 3. Let $x$ be the horizontal distance of the balloon from the observer. We are given $\frac{dx}{dt} = 1$ m/s. 4. Let $z$ be the direct distance from the observer to the balloon. We need to find $\frac{dz}{dt}$. 5. The relationship between $x, h,$ and $z$ is given by the Pythagorean theorem: $x^2 + h^2 = z^2$. 6. Differentiate this equation with respect to time $t$ to relate the rates. 7. Find the values of $x$ and $z$ at the specific instant ($t=40$s) and substitute them to find $\frac{dz}{dt}$.
Step 3: Detailed Explanation
1. Set up the geometric model: Let the observer be at the origin (0,0). The balloon's altitude is constant at 30 m. Let its horizontal position be $x$. The position of the balloon is $(x, 30)$. The direct distance from the observer to the balloon is $z$. By the Pythagorean theorem: \[ x^2 + 30^2 = z^2 \] \[ x^2 + 900 = z^2 \] 2. Differentiate with respect to time: Differentiating both sides of the equation with respect to time $t$: \[ \frac{d}{dt}(x^2 + 900) = \frac{d}{dt}(z^2) \] \[ 2x \frac{dx}{dt} + 0 = 2z \frac{dz}{dt} \] \[ x \frac{dx}{dt} = z \frac{dz}{dt} \] Solving for $\frac{dz}{dt}$: \[ \frac{dz}{dt} = \frac{x}{z} \frac{dx}{dt} \] 3. Find values at t = 40s: We are given $\frac{dx}{dt} = 1$ m/s. Assuming the balloon starts its horizontal motion from a point directly above the observer (i.e., $x=0$ at $t=0$), after 40 seconds: \[ x = \text{speed} \times \text{time} = 1 \, \text{m/s} \times 40 \, \text{s} = 40 \, \text{m} \] Now, find the direct distance $z$ at this instant using the Pythagorean theorem: \[ z = \sqrt{x^2 + 30^2} = \sqrt{40^2 + 30^2} = \sqrt{1600 + 900} = \sqrt{2500} = 50 \, \text{m} \] 4. Calculate $\frac{dz{dt}$:} Substitute the values of $x, z,$ and $\frac{dx}{dt}$ into the related rates equation: \[ \frac{dz}{dt} = \frac{40}{50} \cdot (1) = \frac{4}{5} = 0.8 \, \text{m/s} \] Step 4: Final Answer
The rate at which the balloon is moving away directly from the observer at the 40th second is 0.8 m/s.