A football player kicks a ball with an initial upward velocity of 60 feet/s. The situation is modeled by h=-16t^2+60t+0 where h is Hight and t is time is seconds. What is the maximum range that the ball will reach? I do not understand how to do this. Thanks for your help.

Answer 1

To find the maximum height that the football will reach, you can analyze the quadratic equation given:

\[ h(t) = -16t^2 + 60t + 0 \]

This equation is a standard form of a quadratic equation \( h(t) = at^2 + bt + c \), where:

• \( a = -16 \)
• \( b = 60 \)
• \( c = 0 \)

Step 1: Find the time at which the maximum height occurs

The maximum height of a parabola can be found using the formula:

\[ t = -\frac{b}{2a} \]

Substituting the values of \( a \) and \( b \):

\[ t = -\frac{60}{2 \times -16} = \frac{60}{32} = \frac{15}{8} \text{ seconds} \]

Step 2: Find the maximum height

Now, substitute \( t = \frac{15}{8} \) back into the height equation to find \( h(t) \):

\[ h\left(\frac{15}{8}\right) = -16\left(\frac{15}{8}\right)^2 + 60\left(\frac{15}{8}\right) + 0 \]

Calculating \( \left(\frac{15}{8}\right)^2 \):

\[ \left(\frac{15}{8}\right)^2 = \frac{225}{64} \]

Now, substituting this back into the equation:

\[ h\left(\frac{15}{8}\right) = -16 \cdot \frac{225}{64} + 60 \cdot \frac{15}{8} \] Calculating each term:

1. \( -16 \cdot \frac{225}{64} = -\frac{3600}{64} = -\frac{225}{4} \)
2. \( 60 \cdot \frac{15}{8} = \frac{900}{8} = 112.5 \)

Now putting it all together:

\[ h\left(\frac{15}{8}\right) = -\frac{225}{4} + 112.5 \] Converting \( 112.5 \) to quarters for easier addition:

\[ 112.5 = \frac{450}{4} \]

Now combine:

\[ h\left(\frac{15}{8}\right) = -\frac{225}{4} + \frac{450}{4} = \frac{225}{4} \]

Final Answer

Thus, the maximum height the ball will reach is:

\[ h = \frac{225}{4} \text{ feet} \approx 56.25 \text{ feet} \]

So, the maximum height that the ball will reach is 56.25 feet.

Answer 2

Thanks. This is difficult for me.

Answer 3

You're welcome! Understanding physics and mathematics concepts can be challenging at times, but it's great that you're seeking help. If you find any part difficult or have more questions about this problem or similar topics, feel free to ask! Here’s a quick summary of the steps we took in finding the maximum height for clarity:

1. Identify the equation: You started with the quadratic equation for height, \( h(t) = -16t^2 + 60t \).

2. Find the time for maximum height: Use the formula \( t = -\frac{b}{2a} \) to find the time at which the maximum height occurs.

3. Calculate the maximum height: Substitute that time back into the height equation to find the maximum height.

Remember, tackling problems step by step can make it easier, and practicing different problems will help you get more comfortable with these concepts. Just take your time, and don't hesitate to reach out if you need further assistance!

Answer 4

Nora is creating a dog run for her dog. She has enough fencing material for a fence with a perimeter of 120 feet. Situation modeled A= - w^2+60w where A is area and w is width. Using this equation identify maximum range or area the fence can have. Answer in sq. feet.
Information I know is Perimeter of 120 feet

Answer 5

To find the maximum area that Nora can achieve with a rectangular dog run (considering the perimeter of 120 feet), we first need to understand the relationship between the width \( w \) and the length \( l \) of the rectangle.

1. Connecting Perimeter and Length

The perimeter \( P \) of a rectangle is given by:

\[ P = 2l + 2w \]

Given that \( P = 120 \) feet:

\[ 2l + 2w = 120 \]

Dividing the entire equation by 2:

\[ l + w = 60 \]

From this, we can express the length \( l \) in terms of the width \( w \):

\[ l = 60 - w \]

2. Area in Terms of Width

The area \( A \) of a rectangle is given by:

\[ A = l \times w \]

Substituting \( l = 60 - w \):

\[ A = w(60 - w) = 60w - w^2 \]

This equation matches the form you've provided, \( A = -w^2 + 60w \), which is already set for maximization.

3. Finding Maximum Area

To find the maximum area, we will use the vertex formula for a quadratic equation, \( A = -w^2 + 60w \), where the vertex (maximum point) occurs at:

\[ w = -\frac{b}{2a} \]

In this case, \( a = -1 \) and \( b = 60 \):

\[ w = -\frac{60}{2 \times -1} = \frac{60}{2} = 30 \]

4. Calculating the Length

Now that we have the width, we can find the corresponding length:

\[ l = 60 - w = 60 - 30 = 30 \]

5. Calculating the Maximum Area

Now we can calculate the area:

\[ A = l \times w = 30 \times 30 = 900 \text{ sq. feet} \]

Final Answer

The maximum area that the dog run can have is 900 square feet.