A person standing close to the edge on the top of an 80-foot tower throws a ball with an initial speed of 64 feet per second. After t seconds, the height of the ball above the ground is
s(t) = -16t2 +64t + 80
a. After how many seconds will the ball reach its maximum height?
b. How long will it take before the ball reaches the ground?
c. What is the maximum height of the ball?
you will need to find the vertex of this parabola, a) and c) can then be answered
the t of the vertex is -b/(2a)
= -64/-32 = 2
s(2) -16(4) + 64(2) + 80 = 144
the vertex is (2,144), what does that vertex tell you?
b) when it reaches the ground, s = 0
-16t^2 + 64t + 80 = 0
divide be -16
t^2 - 4t - 5 = 0
(t-5)(t+1) = 0
t = 5 or t is a negative
it will 5 seconds to hit the ground
a. After how many seconds will the ball reach its maximum height?
Well, in order to find the maximum height, we first need to find the vertex of the quadratic equation. You know what they say, "It's all about the vertex, 'bout the vertex, no troubles!"
The equation for the height of the ball, s(t), is given as s(t) = -16t^2 + 64t + 80. To find the time when the ball reaches its maximum height, we can use the formula t = -b/2a. In this case, a = -16 and b = 64.
So, plugging these values into the formula, we have t = -64/(2*(-16)). Neglecting the negative values for time (because, seriously, who wants to deal with math in the negatives?), we find t = 2 seconds. So after two seconds, the ball reaches its maximum height.
b. How long will it take before the ball reaches the ground?
Now that's a tough question. The ball is probably thinking, "Do I really have to come back down to Earth? Can't I just stay up in the air forever?" Well, Mr. Ball, unfortunately, gravity has other plans for you.
To find out how long it takes for the ball to hit the ground, we need to solve the equation s(t) = 0. In this case, our equation is -16t^2 + 64t + 80 = 0. We can either factor it (if we're feeling fancy), or we can use the quadratic formula, which is -b±√(b^2-4ac)/2a. Trust me, it's not as scary as it sounds.
After some calculations, we find that the ball will hit the ground after approximately 5 seconds. So, you better watch out if you're standing beneath that tower!
c. What is the maximum height of the ball?
Ah, the pinnacle of this ball's journey. The highest point it will reach before succumbing to gravity's undeniable pull. The maximum height of the ball can be found by plugging the value of t (2 seconds) into the equation s(t) = -16t^2 + 64t + 80.
Calculating this, we find that the maximum height of the ball is 144 feet. So, for a brief moment, the ball gets a taste of what it's like to be on top of the world before being brought back down to reality. Life is full of ups and downs, my friend!
To find the answers to the questions, we'll need to analyze the given equation for the height of the ball above the ground:
s(t) = -16t^2 + 64t + 80
a) To find the time at which the ball reaches its maximum height, we need to determine the vertex of the parabolic equation. The time of the vertex can be found using the formula:
t = -b / (2a)
For our equation, a = -16 and b = 64. Plugging in these values:
t = -(64) / (2(-16))
t = -64 / -32
t = 2
Therefore, the ball will reach its maximum height after 2 seconds.
b) To determine how long it takes for the ball to reach the ground, we need to find the time when the height, s(t), equals zero. Setting s(t) equal to zero and solving for t:
-16t^2 + 64t + 80 = 0
Dividing through by -8 to simplify the equation:
2t^2 - 8t - 10 = 0
Using the quadratic formula: t = (-b ± √(b^2 - 4ac)) / (2a)
t = (-(-8) ± √((-8)^2 - 4(2)(-10))) / (2(2))
t = (8 ± √(64 + 80)) / 4
t = (8 ± √144) / 4
t = (8 ± 12) / 4
This results in two possible values: t = (8 + 12) / 4 = 5 and t = (8 - 12) / 4 = -1/2. Since time cannot be negative in this situation, we discard the negative value.
Therefore, it will take 5 seconds for the ball to reach the ground.
c) The maximum height of the ball can be found by substituting the value of t = 2 into the equation s(t) = -16t^2 + 64t + 80:
s(2) = -16(2)^2 + 64(2) + 80
s(2) = -16(4) + 128 + 80
s(2) = -64 + 128 + 80
s(2) = 144
Thus, the maximum height of the ball is 144 feet.
To answer these questions, we need to analyze the given quadratic function s(t) = -16t^2 + 64t + 80, which represents the height of the ball above the ground at time t.
a. To find the maximum height of the ball, we need to determine when the vertex of the parabolic function occurs. The vertex of a quadratic function can be found using the formula t = -b / (2a), where a and b are the coefficients of the quadratic equation in standard form (s(t) = at^2 + bt + c).
In our case, a = -16 and b = 64. Plugging these values into the equation, we have:
t = -64 / (2 * -16)
t = -64 / -32
t = 2
Therefore, the ball will reach its maximum height after 2 seconds.
b. To determine how long it will take for the ball to reach the ground, we need to find the time when the height of the ball is equal to zero. This represents the point at which the ball hits the ground.
Setting s(t) = 0 in the equation -16t^2 + 64t + 80 = 0, we can solve for t by factoring or using the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values a = -16, b = 64, and c = 80, we have:
t = ( -64 ± √((64 ^2) - (4 * -16 * 80)) ) / (2 * -16)
After simplifying the equation, we get two values for t: t = 2 and t = 5.
Since we are looking for the time it takes before the ball reaches the ground, we only consider the positive value: t = 5.
Therefore, it will take 5 seconds for the ball to reach the ground.
c. To find the maximum height of the ball, we substitute t = 2 into the equation s(t) = -16t^2 + 64t + 80:
s(2) = -16(2)^2 + 64(2) + 80
s(2) = -16(4) + 128 + 80
s(2) = -64 + 128 + 80
s(2) = 144
So, the maximum height of the ball is 144 feet.