Ben throws a ball vertically upward from the top of a cliff. The height of the ball above the base is approximated by the model h = -5t² - 5t + 100, where h is the height in metres and t is the time in seconds.
a) After how many seconds does the ball hit the ground?
b) How long does it take for the ball to reach a height of 40 metres above the base of the cliff?
a) set -5t² - 5t + 100 = 0 and solve for t
hint: divided all terms by -5, and then it factors nicely
b) -5t² - 5t + 100 = 40
solve as a quadratic using your favourite method
To find out when the ball hits the ground, we need to determine the value of t when h is equal to 0. So, let's set up the equation:
0 = -5t² - 5t + 100
This is a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± √(b² - 4ac)) / 2a
For our equation, a = -5, b = -5, and c = 100. Plugging these values into the quadratic formula, we get:
t = (-(-5) ± √((-5)² - 4(-5)(100))) / (2(-5))
t = (5 ± √(25 + 2000)) / -10
t = (5 ± √2025) / -10
Now, taking the square root of 2025, we get:
t = (5 ± 45) / -10
Simplifying further, we have:
t = (5 + 45) / -10 = 50 / -10 = -5
t = (5 - 45) / -10 = -40 / -10 = 4
Since time cannot be negative, we disregard t = -5. Therefore, the ball hits the ground after 4 seconds.
Now, let's find out how long it takes for the ball to reach a height of 40 meters above the base of the cliff. We can set up the equation:
40 = -5t² - 5t + 100
Simplifying, we have:
0 = -5t² - 5t + 60
This is also a quadratic equation. Again, we can solve it using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula again:
t = (-(-5) ± √((-5)² - 4(-5)(60))) / (2(-5))
t = (5 ± √(25 + 1200)) / -10
t = (5 ± √1225) / -10
Taking the square root of 1225, we get:
t = (5 ± 35) / -10
Simplifying further:
t = (5 + 35) / -10 = 40 / -10 = -4
t = (5 - 35) / -10 = -30 / -10 = 3
Again, we disregard t = -4 since time cannot be negative. Therefore, it takes 3 seconds for the ball to reach a height of 40 meters above the base of the cliff.
To find the answers to these questions, we need to solve the given equations for different values of height (h) and time (t). Let's solve them step by step:
a) To find the time it takes for the ball to hit the ground, we need to find the value of t when the height (h) is equal to zero. By substituting h = 0 into the equation:
0 = -5t² - 5t + 100
This equation is a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
The quadratic formula states that for an equation in the form ax² + bx + c = 0, the values of x can be found using the formula:
x = (-b ± √(b² - 4ac)) / (2a)
In our equation, a = -5, b = -5, and c = 100. Plugging these values into the quadratic formula, we get:
t = (-(-5) ± √((-5)² - 4(-5)(100))) / (2(-5))
Simplifying further:
t = (5 ± √(25 + 2000)) / (-10)
t = (5 ± √2025) / (-10)
t = (5 ± 45) / (-10)
This gives us two possible solutions:
t₁ = (5 + 45) / (-10) = 50 / (-10) = -5
t₂ = (5 - 45) / (-10) = -40 / (-10) = 4
Since time cannot be negative in this context, we discard the negative value for t. Therefore, the ball hits the ground after 4 seconds.
b) To find how long it takes for the ball to reach a height of 40 meters, we need to find the value of t when the height (h) is equal to 40. By substituting h = 40 into the equation:
40 = -5t² - 5t + 100
We have another quadratic equation. Using the same steps as above, we can solve for t:
-5t² - 5t + 100 - 40 = 0
-5t² - 5t + 60 = 0
Factoring this equation, we find:
(-5t + 10)(t - 6) = 0
Setting each factor equal to zero:
-5t + 10 = 0 --> t = 10/5 = 2
t - 6 = 0 --> t = 6
Again, we discard the negative value. Therefore, it takes the ball 2 seconds to reach a height of 40 meters.
So, the answers to the questions are:
a) The ball hits the ground after 4 seconds.
b) It takes 2 seconds for the ball to reach a height of 40 meters above the base of the cliff.