Problem 73 - Algebra and Trigonometry with Analytic Geometry

Question

A hot-air balloon is released at 1:00 P.M. and rises vertically at a rate of \(2 \mathrm{~m} / \mathrm{sec}\). An observation point is situated 100 meters from a point on the ground directly below the balloon (see the figure). If \(t\) denotes the time (in seconds) after 1:00 P.M., express the distance \(d\) between the balloon and the observation point as a function of \(t\).

Step-by-Step Solution

Answer

The distance function is \( d(t) = \sqrt{4t^2 + 10000} \).

Step 1: Understand the Problem

We are tasked with expressing the distance between a hot-air balloon, which ascends vertically, and an observation point on the ground. The observation point is located 100 meters from where the balloon is directly above the ground. We need to find a function for the distance as a function of time, denoted as \( t \).

Step 2: Determine the Height of the Balloon

The balloon rises at a rate of \( 2 \text{ m/sec} \). Therefore, the height of the balloon after \( t \) seconds will be \( 2t \) meters.

Step 3: Use the Pythagorean Theorem

The situation forms a right triangle where the vertical distance from the ground to the balloon is one leg, the horizontal distance from the observation point to the point directly below the balloon is the other leg, and the hypotenuse is the distance \( d \).

Step 4: Express Distance \( d \) Using the Pythagorean Theorem

Using the Pythagorean Theorem, which states \( a^2 + b^2 = c^2 \), we have:\[(2t)^2 + 100^2 = d^2\]Simplifying, we find:\[4t^2 + 10000 = d^2\]

Step 5: Solve for \( d \)

To find \( d \), take the square root of both sides:\[d = \sqrt{4t^2 + 10000}\]

Step 6: Conclude with the Function

Thus, the distance \( d \) as a function of time \( t \) is given by:\[d(t) = \sqrt{4t^2 + 10000}\]

Key Concepts

Right Triangle in the Balloon Scenario

In this scenario, a right triangle helps us understand the connection between the height of the hot-air balloon and its horizontal distance from the observation point. A right triangle has one angle equal to 90 degrees. It consists of three sides: the two legs and the hypotenuse, the longest side.
The vertical rise of the balloon forms one leg of the triangle. This leg changes with time as the balloon goes up.
The fixed ground distance of 100 meters from the observation point forms the other leg.
The hypotenuse, which we need to calculate, represents the slant distance from the observation point to the balloon. The properties of a right triangle, specifically the Pythagorean Theorem, are key to solving this problem.

Understanding the Distance Function

The distance function in this context determines how far the observation point is from the hot-air balloon at any given time. We express this distance as a mathematical function of time, denoted as \(d(t)\).
Because the balloon rises vertically, increasing its height over time, the resulting distance function is dynamic.
To find this function, we combine the constant 100-meter horizontal distance with the balloon's height, using the Pythagorean Theorem.

• Balloon height after \( t \) seconds: \(2t\) meters.
• Pythagorean Theorem for distance \(d\): \((2t)^2 + 100^2 = d^2\).

Simplifying gives us the function \(d(t) = \sqrt{4t^2 + 10000}\).
Here, \(d(t)\) tells us the distance at any time \(t\).

Exploring the Rate of Change

The rate of change describes how quickly one quantity changes relative to another. In our problem, the balloon ascends at a steady rate of \(2 \text{ m/sec}\).
This constant rate tells us exactly how much the balloon rises per second.
It is crucial to understanding how the height of the balloon alters over time. With each passing second, the height of the balloon increases by \(2 \text{ meters}\).

• Height at time \(t\): \(2t\).
• The rate creates a linear relationship between time and height.

The rate of rise is at the core of expressing the balloon's height as \(2t\) in our equation. It also impacts the overall change in distance \(d\), making rate an essential component of our analysis.

Algebraic Expression of the Scenario

Algebraic expressions allow us to model real-world scenarios with mathematical language. In our hot-air balloon problem, an algebraic expression helps describe the distance \(d\) as a function of time \(t\).
To derive the expression for \(d\), we use known quantities to set up an equation based on the Pythagorean Theorem.
This gives us: \((2t)^2 + 100^2 = d^2\). After simplifying:

• Height term: \((2t)^2 = 4t^2\).
• Constant horizontal distance: \(100^2 = 10000\).
• Final expression: \(d(t) = \sqrt{4t^2 + 10000}\).

In this expression, all components, including the square and square root, are expressed algebraically to solve for the variable \(d\).
This showcases how algebra is pivotal in translating physical reality into a computable form.