Question
The position of a ball after it is kicked can be determined by using the function f(x)=−0.11x2+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 find the height of the ball when it is kicked, we need to evaluate the function \( f(x) \) at the point where the horizontal distance \( x \) is 0:
\[
f(0) = -0.11(0)^2 + 2.2(0) + 1 = 1
\]
So, the height of the ball when it is kicked is **1 foot**.
Next, to find the highest point of the ball in the air, we need to determine the vertex of the parabola given by the function \( f(x) = -0.11x^2 + 2.2x + 1 \). The vertex \( x \)-coordinate can be found using the formula:
\[
x = -\frac{b}{2a}
\]
where \( a = -0.11 \) and \( b = 2.2 \). Thus we have:
\[
x = -\frac{2.2}{2 \times -0.11} = -\frac{2.2}{-0.22} = 10
\]
Now we substitute \( x = 10 \) back into the function to find the corresponding height:
\[
f(10) = -0.11(10)^2 + 2.2(10) + 1
\]
Calculating this step-by-step:
1. \( (10)^2 = 100 \)
2. \( -0.11 \times 100 = -11 \)
3. \( 2.2 \times 10 = 22 \)
Now substituting these values into the equation:
\[
f(10) = -11 + 22 + 1 = 12
\]
Thus, the highest point of the ball in the air is **12 feet**.
In summary:
- The height when kicked: 1 foot
- The highest point of the ball: 12 feet
\[
f(0) = -0.11(0)^2 + 2.2(0) + 1 = 1
\]
So, the height of the ball when it is kicked is **1 foot**.
Next, to find the highest point of the ball in the air, we need to determine the vertex of the parabola given by the function \( f(x) = -0.11x^2 + 2.2x + 1 \). The vertex \( x \)-coordinate can be found using the formula:
\[
x = -\frac{b}{2a}
\]
where \( a = -0.11 \) and \( b = 2.2 \). Thus we have:
\[
x = -\frac{2.2}{2 \times -0.11} = -\frac{2.2}{-0.22} = 10
\]
Now we substitute \( x = 10 \) back into the function to find the corresponding height:
\[
f(10) = -0.11(10)^2 + 2.2(10) + 1
\]
Calculating this step-by-step:
1. \( (10)^2 = 100 \)
2. \( -0.11 \times 100 = -11 \)
3. \( 2.2 \times 10 = 22 \)
Now substituting these values into the equation:
\[
f(10) = -11 + 22 + 1 = 12
\]
Thus, the highest point of the ball in the air is **12 feet**.
In summary:
- The height when kicked: 1 foot
- The highest point of the ball: 12 feet