Question 1149613: The path of the longest shot put by the women's track team at Sun Devil U is modeled by h(x)=-0.017x^2+1.09x+6.1, where x represents the horizontal distance from the start and h(x) is the height of the shot put above ground. (Both x and h(x) are measured in feet)
a. Determine h(40). Round your answer to 2 decimal places. Then explain what your answer means in the context of the problem.
b. Determine the numerical value of the vertical intercept and explain what this means in the context of the problem.
c. Determine the numerical values of the vertex intercept and explain what they mean in the context of the problem.
d. How far from the start did the shot put strike the ground? Round your answer to 2 decimal places.
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Part A
Plug in x = 40. This means we replace every copy of x with 40, then we use PEMDAS to simplify. Use of a calculator makes quick work of this.
Recall that x is defined to be the horizontal distance from the start, and h(x) is the corresponding height of the shot put that pairs with x.
We have h(40) = 22.5 which means x = 40 and h(x) = 22.5 pair up together.
Therefore, when the shot put is 40 horizontal feet away from the starting point, the item is also 22.5 feet off the ground.
This point is located at (40, 22.5) which is marked as point A in the diagram below.
=======================================================
Part B
Plug in x = 0 to find the y intercept, which is the vertical intercept. This is the point where the parabola crosses the y axis.
The starting point is 6.1 feet off the ground.
This point is located at (0, 6.1) which is marked as point B in the diagram below.
=======================================================
Part C
I think you meant to ask about the vertex, instead of the "vertex intercept"?
If so, then compare the form  to the equation  . We see that a,b,c are:
a = -0.017
b = 1.09
c = 6.1
The goal is to find the vertex (h,k). First let's find h.
 Rounding to 2 decimal places
The x coordinate of the vertex is approximately 32.06
---------
Plug that x value into the h(x) function to find the y coordinate of the vertex
 Rounding to 2 decimal places
The value of k is roughly 23.57
The vertex is approximately located at (32.06, 23.57). This is marked as point C in the diagram below.
---------
Interpretation: The shot put is at its highest point at the vertex mentioned above. This is because the shot put starts going up, then gravity pulls it back down until it hits the ground.
The shot put's max height is about 23.57 feet, which occurs when it is roughly 32.06 horizontal feet away from the starting point.
=======================================================
Part D
The shot put strikes the ground when y = 0 or h(x) = 0; i.e, when the height is 0
 y is replaced with 0
To solve this equation, we can use the quadratic formula
Plug in a = -0.017, b = 1.09, and c = 6.1
 or 
 or 
 or 
 or 
 or 
 or 
 or  Rounding to 2 decimal places
These values are the two approximate x intercepts.
Alternatively, you can use your graphing calculator's root finder to locate the two x intercepts.
I'm using the graphing calculator GeoGebra to help find the roots, and to also generate the image below.
The roots that GeoGebra generates matches up with what the quadratic formula produces.
Points D and E are the two roots, or x intercepts
We will ignore point E as this has a negative x coordinate. A negative distance away makes no sense. Also, the shot put starts at point B and moves to the right until it lands on the ground at point D. This leaves point E out entirely.
When the shot put hits the ground, the object lands roughly 69.30 horizontal feet away from the starting point.
|
|
|
