Question

Due to Earth's curvature, the maximum distance, \(d\), in kilometers that a person can see from a height \(h\) meters above the ground is given by the formula $$d=\frac{7}{2} \sqrt{h}$$ Use the formula to determine the maximum distance a person can see from a height of \(20 \frac{1}{4}\) meters.

Short answer

Answer: The maximum distance a person can see from a height of \(20 \frac{1}{4}\) meters is \(\frac{63}{4}\) kilometers.

Step-by-step solution

Convert the mixed number to an improper fraction.

First, we need to convert the mixed number \(20 \frac{1}{4}\) to an improper fraction, so it's easier to work with. To do this, we multiply the whole number (20) by the denominator (4), and then add the numerator (1). This will give us the new numerator. The denominator remains the same. $$20 \frac{1}{4} = \frac{20 \times 4 + 1}{4}$$

Calculate the new numerator for the improper fraction.

Now we need to perform the calculation within the brackets to find the new numerator: $$20 \times 4 + 1 = 80 + 1 = 81$$ So our improper fraction is: $$20 \frac{1}{4} = \frac{81}{4}$$

Substitute the value of \(h\) into the given formula.

Since \(h = 20 \frac{1}{4}\) meters, we can write it as \(h = \frac{81}{4}\) meters. Now we plug in this value of \(h\) in the formula for maximum distance: $$d = \frac{7}{2} \sqrt{\frac{81}{4}}$$

Simplify the square root term.

Simplify the square root term inside the formula: $$\sqrt{\frac{81}{4}} = \frac{\sqrt{81}}{\sqrt{4}} = \frac{9}{2}$$

Calculate the final distance d.

Now that we have simplified the square root term, we can multiply it by the constant \(\frac{7}{2}\) and find the maximum distance: $$d = \frac{7}{2} \times \frac{9}{2} = \frac{63}{4}$$ The maximum distance a person can see from a height of \(20 \frac{1}{4}\) meters is \(\frac{63}{4}\) kilometers.