during a match, pat kicks a football onto the roof of the stand. The stand and the path of the ball are at righ angles to each other.
The path of the ball is given by y=2.5x-x^2÷15
The roof of the stand is given by y=x÷2+48 ÷5, where 18</=x</=40. All demensions are in metres. The origin is where pat kicks the ball from.
1.Find the height of the ball when it hits the roof of the stand for the first time.
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To find the height of the ball when it hits the roof of the stand for the first time, we need to find the coordinates where the path of the ball intersects with the roof of the stand.
First, let's find the coordinates where the path of the ball intersects with the roof of the stand by setting the equations equal to each other:
2.5x - x^2/15 = x/2 + 48/5
Next, we simplify and rearrange the equation to bring all the terms to one side:
2.5x - x^2/15 - x/2 - 48/5 = 0
Multiply the equation by 30 to eliminate the fractions:
75x - 2x^2 - 15x - 288/5 = 0
Combine like terms:
-2x^2 + 60x - 288/5 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = -2, b = 60, and c = -288/5. Substituting these values into the quadratic formula:
x = (-60 ± √(60^2 - 4(-2)(-288/5))) / (2(-2))
Next, simplify:
x = (-60 ± √(3600 + 2304/5)) / (-4)
x = (-60 ± √(3600 + 460.8)) / (-4)
x = (-60 ± √(4060.8)) / (-4)
x = (-60 ± √4060.8) / (-4)
Now, we have two possible solutions for x. We need to determine which one falls within the given range of 18 ≤ x ≤ 40.
Calculating the solutions:
x1 = (-60 + √4060.8) / (-4)
x2 = (-60 - √4060.8) / (-4)
Using a calculator, we find:
x1 ≈ 32.39
x2 ≈ -2.39
Since x2 is not within the given range, we can disregard it.
Now, plug the value of x1 back into either of the original equations to find the corresponding y-coordinate:
Using y = 2.5x - x^2/15, substitute x = 32.39:
y ≈ 2.5(32.39) - (32.39^2)/15
y ≈ 80.975 - 223.159
y ≈ -142.184
Since we are measuring height, we disregard the negative value and take the absolute value of y:
The height of the ball when it hits the roof of the stand for the first time is approximately 142.184 meters.