David kicks a ball off the ground. After t seconds, it’s height, h (in feet), is given by the formula h=-16t^2+80t. How many sec
1 answer:
Answer:
<em>The ball reaches its maximum height at 2.5 seconds</em>
<em>The maximum height is 100 feet</em>
Step-by-step explanation:
<u>Maximum Value of Functions</u>
We use the derivative of a function to find its maximum or minimum value over a given interval.
Given a function y=f(x), the first derivative criterion establishes if x=a is such that:
f'(a)=0, and f''(a) < 0 then x=a is a maximum of f.
The height h in feet of a ball after t seconds is:
Find the first derivative:
Equate to 0:
Subtract 80:
Divide by -32:
t=2.5 seconds
Find the second derivative:
h''(t)=-32
Since h''(t) is always negative, then
The ball reaches its maximum height at 2.5 seconds
To find the value of the maximum height, substitute t=2.5 into the function:
The maximum height is 100 feet
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where a and b are the coefficients of the function. For f(x), we have a = -1 and b = -16. Thus, we have
Since we now have the x-coordinate of the vertex, we can just easily substitute 8 into f(x) as shown below.
Thus, the vertex of f(x) is (8, 60). The vertex of the function shows the maximum value that the function can reach. For this particular case, the vertex represents the maximum profit that the shop owner could earn.
The x-intercepts of the function are the values of x when f(x) is zero. By equation f(x) with zero, we can solve the quadratic equation to find the x-intercepts as shown below.
From this, we can see that the x-intercepts are (6, 0) and (10, 0). We just discussed that the x-intercepts are the values of x when y is zero. So, for this case it means that the shop does not earn a profit if they sell 6 or 10 candles. Basically, the x-intercepts represent the number candles sold if the shop wants to break even.