Problem 80 - Algebra A Combined Function

Question

The maximum distance \(D(h)\) in kilometers that a person can see from a height h kilometers above the ground is given by the function \(D(h)=111.7 \sqrt{h}\). Round your answers to two decimal places. Find the height that would allow a person to see 40 kilometers.

Step-by-Step Solution

Answer

The height is approximately 0.13 kilometers.

Step 1: Understanding the Problem

We are given the formula for the maximum distance a person can see from a height above the ground, which is \(D(h) = 111.7 \sqrt{h}\). We need to find the height \(h\) at which the distance \(D(h)\) is 40 kilometers.

Step 2: Set Up the Equation

Since we are looking for the height \(h\) when the distance \(D(h)\) is 40 kilometers, we set up the equation: \[ 111.7 \sqrt{h} = 40 \]

Step 3: Solve for \(\sqrt{h}\)

Divide both sides of the equation by 111.7 to isolate \(\sqrt{h}\): \[ \sqrt{h} = \frac{40}{111.7} \] Calculate the right side to find \(\sqrt{h}\): \[ \sqrt{h} \approx 0.358 \]

Step 4: Solve for \(h\)

Square both sides of the equation \(\sqrt{h} \approx 0.358\) to solve for \(h\): \[ h = (0.358)^2 \] Calculate \(h\): \[ h \approx 0.128 \]

Step 5: Round to Two Decimal Places

Round \(h = 0.128\) to two decimal places:\[ h \approx 0.13 \]

Key Concepts

Maximum Distance Calculation

Understanding how to calculate maximum distance using algebra functions is a valuable skill. To calculate the maximum distance a person can see from a certain height, you can use the formula \(D(h) = 111.7 \sqrt{h}\).
This equation tells us that the distance \(D\) is proportional to the square root of the height \(h\) from which the person is observing. Here, 111.7 is a constant that relates height and distance. It indicates how far you can see in terms of kilometers when looking from a specific height above the ground.
To find maximum visible distance or height, it's important to understand the relationship that this formula is depicting. This formula helps to predict how high you need to be to see a certain kilometer distance. Applying this knowledge allows us to perform useful calculations in areas such as navigation, aviation, and even for recreational activities like hiking.
Whenever you're solving such problems, remember to work through units thoroughly. Distance and height should be in the same units (kilometers in this case) to ensure accurate calculations.

Squaring Method

One key mathematical technique in solving our original exercise involves the squaring method. This method is crucial when dealing with equations that include a square root, such as \(\sqrt{h}\).
To "square" something means to multiply a number by itself. When we square both sides of an equation, we're essentially removing the square root sign. Let's go through this step-by-step:

• Starting with the equation \(\sqrt{h} = 0.358\).
• To eliminate the square root, square both sides: \(h = (0.358)^2\).
• This results in \(h = 0.128\).

Squaring is useful because it helps transition from a radical expression (one involving a square root) to a simpler one that can be solved more easily. However, always remember that when you square terms, especially on equations that might be more complex, it’s important to check your final answers against the original problem. This ensures that squaring didn't introduce extraneous solutions that don’t actually satisfy the original equation.

Solving Equations

Solving equations is a fundamental process in algebra functions, especially when finding unknown values, like height in our original exercise. The goal when solving equations is to isolate the variable you're solving for, step by step.
Let’s see how we can approach solving the equation \(111.7 \sqrt{h} = 40\).

• First, you set up the equation properly based on what you need to find. Here, we want to find the height \(h\).
• Next, isolate \(\sqrt{h}\) by performing inverse operations. Divide both sides by 111.7: \(\sqrt{h} = \frac{40}{111.7}\).
• Calculate this to simplify the right side: \(\sqrt{h} \approx 0.358\).
• Since \(\sqrt{h}\) is isolated, square both sides to solve for \(h\).

Each step brings you closer to finding the unknown variable accurately. Pay careful attention to the operations you choose, and double-check each step to verify accuracy. Solving equations involves not only finding a result but also ensuring that the result fits within the constraints of the original problem, like rounding appropriately based on the context.