Question
The position of a ball after it is kicked can be determined by using the function f(x)=−0.11x^2+2.2x+1, where f(x) is the height, in feet, above the ground and x is the horizontal distance, in feet, of the ball from the point at which it was kicked. What is the height of the ball when it is kicked? What is the highest point of the ball in the air?
Answers
GPT-4o mini
To determine the height of the ball when it is kicked and its highest point in the air, we will analyze the function \( f(x) = -0.11x^2 + 2.2x + 1 \).
**Step 1: Height of the ball when it is kicked**
The ball is kicked from the initial point, which corresponds to \( x = 0 \). We can find the height at this point by substituting \( x = 0 \) into the function:
\[
f(0) = -0.11(0)^2 + 2.2(0) + 1 = 1.
\]
So, the height of the ball when it is kicked is **1 foot**.
**Step 2: Highest point of the ball in the air**
The highest point of the parabola represented by the function \( f(x) \) occurs at its vertex. The vertex of a quadratic function in the form \( ax^2 + bx + c \) is given by the formula:
\[
x = -\frac{b}{2a}.
\]
In our function, \( a = -0.11 \) and \( b = 2.2 \). Plugging in these values:
\[
x = -\frac{2.2}{2(-0.11)} = -\frac{2.2}{-0.22} = 10.
\]
Thus, the horizontal distance at which the ball reaches its highest point is \( x = 10 \) feet.
Next, we’ll find the height at this point by substituting \( x = 10 \) back into the function:
\[
f(10) = -0.11(10)^2 + 2.2(10) + 1.
\]
Calculating this step-by-step:
1. \( 10^2 = 100 \)
2. \( -0.11(100) = -11 \)
3. \( 2.2(10) = 22 \)
4. \( f(10) = -11 + 22 + 1 = 12 \).
So, the highest point of the ball in the air is **12 feet**.
**Summary:**
- Height of the ball when it is kicked: **1 foot**
- Highest point of the ball in the air: **12 feet**.
**Step 1: Height of the ball when it is kicked**
The ball is kicked from the initial point, which corresponds to \( x = 0 \). We can find the height at this point by substituting \( x = 0 \) into the function:
\[
f(0) = -0.11(0)^2 + 2.2(0) + 1 = 1.
\]
So, the height of the ball when it is kicked is **1 foot**.
**Step 2: Highest point of the ball in the air**
The highest point of the parabola represented by the function \( f(x) \) occurs at its vertex. The vertex of a quadratic function in the form \( ax^2 + bx + c \) is given by the formula:
\[
x = -\frac{b}{2a}.
\]
In our function, \( a = -0.11 \) and \( b = 2.2 \). Plugging in these values:
\[
x = -\frac{2.2}{2(-0.11)} = -\frac{2.2}{-0.22} = 10.
\]
Thus, the horizontal distance at which the ball reaches its highest point is \( x = 10 \) feet.
Next, we’ll find the height at this point by substituting \( x = 10 \) back into the function:
\[
f(10) = -0.11(10)^2 + 2.2(10) + 1.
\]
Calculating this step-by-step:
1. \( 10^2 = 100 \)
2. \( -0.11(100) = -11 \)
3. \( 2.2(10) = 22 \)
4. \( f(10) = -11 + 22 + 1 = 12 \).
So, the highest point of the ball in the air is **12 feet**.
**Summary:**
- Height of the ball when it is kicked: **1 foot**
- Highest point of the ball in the air: **12 feet**.