Question

Please help and show work please algebra 2!!! lesson 3.2

8. The height y (in feet) of a javelin (t) seconds after it is thrown can be modeled
by the function y = -16t² +64t+4

a. Find the maximum height of the javelin

b. How long does the javelin take to hit the
ground?

Answer 1

The maximum height of the javelin is 68 feet, reached at 2 seconds after it is thrown. The time it takes for the javelin to hit the ground needs to be calculated by solving the quadratic equation 0 = -16t² +64t+4, yielding the positive value of t as the time of impact.

Step-by-step explanation:

The maximum height of a javelin can be found by completing the square or using the vertex formula for a parabola. Given the quadratic function y = -16t² +64t+4, the vertex occurs at t = -b/(2a), which represents the time at which the javelin reaches its maximum height. Substituting the values of a = -16 and b = 64 into the formula, we find that the time at the maximum height is t = -64/(2*(-16)) = 2 seconds. The maximum height is then found by substituting t = 2 back into the function, yielding y = -16(2)² + 64(2) + 4 = -16(4) + 128 + 4 = 64+4 = 68 feet.

To find the time it takes for the javelin to hit the ground, we need to solve for t when y = 0. Setting the quadratic equation to zero, we get 0 = -16t² +64t+4. This equation can be solved using the quadratic formula, factoring, or graphing. Using the quadratic formula, we find that t = (-b±√(b²-4ac))/(2a). Substituting a = -16, b = 64, and c = 4, we get t = (64±√(64²-4(-16)(4)))/(2(-16)). The positive solution of this equation will give us the time when the javelin hits the ground.

The correct calculations and substitutions will yield the desired time of flight, completing the problem.

Answer 2

Let's solve the problem step by step.

a. Find the maximum height of the javelin:

The given function is y = -16t² + 64t + 4, where y represents the height in feet and t represents the time in seconds.

To find the maximum height of the javelin, we need to determine the vertex of the quadratic function. The vertex of a quadratic function in the form y = ax² + bx + c is given by the formula:

t = -b / (2a)

In this case, a = -16 and b = 64. Let's substitute these values into the formula:

t = -64 / (2 * -16)
t = -64 / -32
t = 2

The time t = 2 seconds represents the moment when the javelin reaches its maximum height.

Now, let's find the maximum height by substituting t = 2 into the given function:

y = -16(2)² + 64(2) + 4
y = -16(4) + 128 + 4
y = -64 + 128 + 4
y = 68

Therefore, the maximum height of the javelin is 68 feet.

b. How long does the javelin take to hit the ground?

To find the time it takes for the javelin to hit the ground, we need to set y = 0 in the given function and solve for t:

0 = -16t² + 64t + 4

This is a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:

t = (-b ± √(b² - 4ac)) / (2a)

In this case, a = -16, b = 64, and c = 4. Let's substitute these values into the quadratic formula:

t = (-64 ± √(64² - 4(-16)(4))) / (2(-16))
t = (-64 ± √(4096 + 256)) / (-32)
t = (-64 ± √(4352)) / (-32)
t = (-64 ± 66) / (-32)

Now, we have two possible solutions:

1. t = (-64 + 66) / (-32)
t = 2 / (-32)
t = -1/16

2. t = (-64 - 66) / (-32)
t = -130 / (-32)
t = 4.0625

Since time cannot be negative in this context, we discard the negative value.

Therefore, the javelin takes approximately 4.0625 seconds to hit the ground.