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A soccer ball is kicked from the ground at an upward velocity of 25 feet per second...

Check the final answer first, then review the worked steps.

Problem

A soccer ball is kicked from the ground at an upward velocity of 25 feet per second. This situation can be represented by the function f(x) = -16x^2 + 25x. Graph the function. Identify the x-intercept(s) and interpret its meaning.

Answer

(1.563, 0)

Step-by-step solution

  1. Identify the function: The problem provides the function that models the height of the soccer ball over time: $f(x) = -16x^2 + 25x$. Here, $f(x)$ represents the height in feet, and $x$ represents the time in seconds after the ball is kicked.
  1. Understand x-intercepts: The x-intercepts are the points where the graph of the function crosses the x-axis. At these points, the value of the function (the height, $f(x)$) is zero. In the context of this problem, the x-intercepts represent the times when the soccer ball is on the ground.
  1. Set the function to zero: To find the x-intercepts, we set $f(x) = 0$: $$ -16x^2 + 25x = 0 $$
  1. Factor the equation: We can factor out a common term of $x$ from both terms in the equation: $$ x(-16x + 25) = 0 $$

5. Solve for x: For the product of two factors to be zero, at least one of the factors must be zero. Therefore, we have two possible cases:
Case 1: $x = 0$
Case 2: $-16x + 25 = 0$

6. Solve the second case: For the second case, we solve for $x$: $$ -16x = -25 $$
$$ x = \frac{-25}{-16} $$
$$ x = \frac{25}{16} $$

  1. Convert to decimal: To make it easier to compare with the options, we convert the fraction to a decimal: $$ x = 1.5625 $$
  1. Identify the x-intercepts: The x-intercepts are the points where $f(x)=0$. We found two values for $x$: $0$ and $1.5625$. Since $x$ represents time, $x=0$ represents the initial moment the ball is kicked from the ground. The other intercept, $x=1.5625$, represents the time when the ball returns to the ground. Therefore, the x-intercepts are $(0, 0)$ and $(1.5625, 0)$.

9. Interpret the x-intercepts:
- The x-intercept $(0, 0)$ means that at time $0$ seconds, the height of the soccer ball is $0$ feet (it is on the ground).
- The x-intercept $(1.5625, 0)$ means that approximately $1.563$ seconds after being kicked, the soccer ball returns to the ground (its height is $0$ feet).

  1. Select the correct option: Based on the interpretation, the option that correctly identifies an x-intercept and its meaning related to the ball returning to the ground is $(1.563, 0)$. The soccer ball will return to the ground $1.563$ seconds after it has been kicked.