A projectile is thrown upward with an initial velocity of 272 feet per second. After t seconds, its height h(t)above the ground is given by the function : h(t)=-16t^2+272t.
a. Determine the projectile's maximum height.
b. Determine how long it takes the projectile to reach its maximum height.
c. How many seconds will it take the projectile to strike the ground after it is thrown?
Since you called it "College Algebra" I will assume you know Calculus.
h'(t) = -32t + 272
b) max height obtained when
-32t+272 = 0
t = 8.5 seconds
a) h(2.5) = -16(8.5)^2 + 272(8.5) = 1156
c) to strike ground, h(t) = 0
-16t^2 + 272t = 0
-16t(t - 17) = 0
t = 0, (at the start) or t = 17
It will hit the ground 17 seconds later.
a. The maximum height of the projectile can be determined by finding the vertex of the quadratic function h(t) = -16t^2 + 272t. The formula for the x-coordinate of the vertex is given by t = -b/2a, where a = -16 and b = 272. Plug these values into the formula:
t = -272/(2*(-16))
t = -272/(-32)
t = 8.5 seconds
To find the maximum height, substitute this value of t back into the original equation:
h(t) = -16(8.5)^2 + 272(8.5)
h(t) = -16(72.25) + 2312
h(t) = -1156 + 2312
h(t) = 1156 feet
Therefore, the projectile's maximum height is 1156 feet.
b. The time it takes for the projectile to reach its maximum height is 8.5 seconds, as determined in part a.
c. To find the time it takes for the projectile to strike the ground after it is thrown, we need to find the positive root of the equation h(t) = -16t^2 + 272t = 0. This represents when the height of the projectile is zero (ground level).
Solve the equation:
-16t^2 + 272t = 0
-16t(t - 17) = 0
From this equation, we have two possible solutions:
t = 0 (when the projectile is initially thrown)
t = 17 (when the projectile strikes the ground)
Since we're looking for the positive root, the time it takes for the projectile to strike the ground after it is thrown is 17 seconds.
a. To determine the projectile's maximum height, we need to find the vertex of the quadratic function h(t) = -16t^2 + 272t.
The vertex of a quadratic function in the form of f(x) = ax^2 + bx + c is given by the formula:
x = -b / (2a)
In this case, a = -16 and b = 272, so substituting these values into the formula, we have:
t = -272 / (2 * -16)
Simplifying further:
t = -272 / -32
t = 8.5
Therefore, the projectile reaches its maximum height at t = 8.5 seconds.
To find the maximum height, substitute t = 8.5 into the function h(t):
h(8.5) = -16(8.5)^2 + 272(8.5)
Simplifying:
h(8.5) = -16(72.25) + 2312
h(8.5) = -1156 + 2312
h(8.5) = 1156
Therefore, the projectile's maximum height is 1156 feet.
b. We already determined that the projectile reaches its maximum height at t = 8.5 seconds.
c. To determine how long it will take for the projectile to strike the ground, we set h(t) = 0 (since the height above the ground is zero when the projectile strikes the ground).
So, we have:
0 = -16t^2 + 272t
Divide the equation by 8 to simplify it:
0 = -2t^2 + 34t
Now we can factor out a common factor:
0 = t(-2t + 34)
Setting each factor equal to zero, we have:
t = 0 or -2t + 34 = 0
Since time cannot be negative, we disregard t = 0.
Solving -2t + 34 = 0:
-2t = -34
Dividing by -2:
t = 17
Therefore, it will take the projectile 17 seconds to strike the ground after it is thrown.
To find the answers to these questions, we will use the given equation h(t) = -16t^2 + 272t.
a. Determine the projectile's maximum height:
The maximum height occurs at the vertex of the parabolic equation. The vertex of a parabola of the form ax^2 + bx + c is given by x = -b / (2a). In this case, a = -16 and b = 272.
x = -272 / (2 * -16) = -272 / -32 = 8.5
To find the maximum height, substitute this value back into the equation:
h(t) = -16(8.5)^2 + 272(8.5)
h(t) = -16(72.25) + 2312
h(t) = -1156 + 2312
h(t) = 1156
Therefore, the maximum height of the projectile is 1156 feet.
b. Determine how long it takes the projectile to reach its maximum height:
The time it takes to reach the maximum height is the x-coordinate of the vertex. In this case, the x-coordinate is 8.5 seconds.
c. How many seconds will it take the projectile to strike the ground after it is thrown?
To find the time it takes for the projectile to strike the ground, we need to find the value of t when h(t) = 0. Setting h(t) equal to zero, we have:
-16t^2 + 272t = 0
Dividing both sides by -16 to simplify:
t^2 - 17t = 0
Factoring out t:
t(t - 17) = 0
Setting each factor equal to zero, we get two possible solutions:
t = 0 or t - 17 = 0
Solving for t, we find that t = 0 or t = 17.
Since we are dealing with time, we discard the t = 0 solution since it represents the initial time when the projectile is thrown.
Therefore, it will take 17 seconds for the projectile to strike the ground after it is thrown.