Would someone check this for me please because I'm a little confused.
The height of a golf ball in meters can be described by the equation h = -4.9t^2 + 22.7t, where t is the number of seconds after it was hit.
a) Find the vertex AND explain what each coordinate means. Round to the nearest tenth if necessary.
So I did vertex = -b/2a -22.7/2(-4.9) which equals 2.3
I plugged that back into the equation -4.9(2.3)^2 + 22.7(2.3) and got 26.3
So my vertex is (2.3, 26.3) which means that 2.3 is the number of seconds after it was hit that it reached it's highest point of 26.3 and I'm going to assume feet since it doesn't say.
b) Find the y-intercept AND explain what it means.
Well, I'm stuck here, but I plugged in 0 for x into the equation since x is zero in your y-intercept. y = -4.9(0)^2 + 22.7(0) and go 0, so my y - intercept would be (0, 0). It means that at zero seconds, before it was hit, it was on the ground.
Very confused and unsure about that one.
c) Find the x-intercepts AND explain what each intercept means.
I used the quadratic formula for this, plugging in -4.9 for a, 22.7 for b, and 0 for c. I got x = 0 and x = 4.6, so my x intercepts would be (0, 0) and (4.6, 0)
I'm guessing here, but think this means that at 0 seconds the ball was on the ground, and then at 4.6 seconds it came back down and hit the ground again.
Would someone please check my work to see if I did this correctly? Thank you.
a) since they give g/2 = -4.9, the units are in meters, not feet.
for feet, g = -32
The statement of the problem even says meters.
b) you are correct. The initial height was 0.
c) also correct
To check your work, let's go step by step through each question.
a) Finding the vertex:
You correctly used the formula for finding the x-coordinate of the vertex, which is -b/2a. However, it seems that there is a calculation error. Let's re-calculate it:
x = -(-22.7) / (2 * (-4.9))
x = 22.7 / 9.8
x ≈ 2.32
So, the x-coordinate of the vertex is approximately 2.32. Now let's find the y-coordinate by substituting this value back into the equation:
h = -4.9 * (2.32)^2 + 22.7 * 2.32
h ≈ 26.3
Therefore, the correct vertex is (2.32, 26.3). This means that the ball reaches its highest point after approximately 2.32 seconds, and the height at that point is approximately 26.3 meters.
b) Finding the y-intercept:
The y-intercept represents the point where the graph intersects the y-axis, which occurs when x = 0. You correctly substituted 0 for x in the equation, and your answer of (0, 0) is indeed correct. This means that at zero seconds (before the ball was hit), the height of the ball is 0 meters, indicating that it is on the ground.
c) Finding the x-intercepts:
The x-intercepts represent the points where the graph intersects the x-axis, which occurs when the height of the ball is 0. You correctly used the quadratic formula to find x, and your solutions x = 0 and x ≈ 4.6 are correct. So the x-intercepts are (0, 0) and (4.6, 0). This means that at t = 0 seconds and t ≈ 4.6 seconds, the ball hits the ground.
Overall, you did a great job with your work and calculations. Just note the slight correction in the vertex calculation.