Problem 106 - College Algebra Essentials

Question

A basketball player's hang time is the time spent in the air when shooting a basket. The formula $$t=\frac{\sqrt{d}}{2}$$ models hang time, \(t,\) in seconds, in terms of the vertical distance of a player's jump, \(d,\) in feet. (image cannot copy) If hang time for a shot by a professional basketball player is 0.85 second, what is the vertical distance of the jump, rounded to the nearest tenth of a foot?

Step-by-Step Solution

Answer

The vertical distance of the jump, rounded to the nearest tenth of a foot, is 2.9 feet.

Step 1: Understand the given formula

The given formula which models the hang time is \(t=\frac{\sqrt{d}}{2}\). This formula means hang time (t) is equal to the square root of the vertical distance (d) divided by 2.

Step 2: Substitute the given value

In this scenario, the hang time (t) of a professional basketball player is provided as 0.85 seconds. Substitute this value into the hang time formula so that 0.85 equals \(\frac{\sqrt{d}}{2}\).

Step 3: Solve for the vertical distance (d)

To solve for d, first multiply both sides of the equation by 2, yielding 1.7 equals \(\sqrt{d}\). Then square both sides to get \(d = 1.7^2\).

Step 4: Calculate the result

Calculate \(d = 1.7^2\) to get the vertical distance. The value should be rounded to the nearest tenth of a foot.

Key Concepts

Hang Time

Hang time is an exciting aspect of basketball, describing the time a player remains airborne during a jump or shot. This metric is crucial for athletes who aim to achieve superior performance and audience appeal.

• Hang time is influenced by the vertical leap of the player.
• The greater the initial jump height, the longer the player can stay in the air.

Hang time not only reflects a player's jumping ability but also contributes to more effective shooting and defensive play in basketball. Understanding the formula that calculates hang time is essential for both athletes aiming to improve and students solving exercises related to this concept.

Square Root

The square root is a fundamental concept in mathematics, often appearing in various equations and expressions. In the context of hang time, it is a pivotal part of the formula.

• A square root of a number is a value that, when multiplied by itself, gives the original number.
• For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9.

Understanding square roots is essential, as they frequently appear in equations involving areas, volumes, and distances. They allow us to reverse quadratic operations and are integral in solving many real-world problems, including those encountered in physical activities like jumping.

Solving Equations

Solving equations is an important skill in mathematics, allowing us to find unknown values within an equation. When we are given a formula, like the hang time equation, solving involves isolating the variable of interest, usually represented by a letter such as \( d \).

• The first step is to substitute the known values into the equation.
• Next, manipulate the equation by performing various operations, such as addition, subtraction, multiplication, or division, to isolate the unknown variable.

In the case of hang time, solving for the vertical distance \( d \) involves using these operations to unravel the equation, revealing the desired measurement.

Substitution Method

The substitution method is a strategy to solve equations and is particularly useful when dealing with formulas involving multiple variables. By substituting known values into an equation, we can progressively solve for unknowns.

• Start by placing the given values into the equation in place of the corresponding variables.
• This simplifies the equation, making it easier to focus on solving for the remaining unspecified variables.

In our hang time example, we substitute the known hang time value \( t = 0.85 \) into the equation \( t = \frac{\sqrt{d}}{2} \). This technique simplifies the problem-solving process, making it more manageable and less prone to error. By using substitution effectively, complex problems become more approachable, unveiling the results step-by-step.