Question

On a windy morning, a hot air balloon starts rising and flying away from the top of a hill. The altitude, h, in feet, of the balloon x hours after starting its ascent from the hill, can be modeled by the function:

h(x) = -16x^2 + 64x + 80.

What is the positive x-intercept of the function h(x)?
a) 2 hours
b) 4 hours
c) 5 hours
d) 6 hours

Explain what the positive x-intercept represents in the given situation.
a) It represents the time when the balloon reaches its maximum altitude.
b) It represents the time when the balloon touches the ground.
c) It represents the time when the balloon reaches its initial starting point.
d) It represents the time when the balloon reaches its maximum speed.

What is the negative x-intercept of the function h(x)?
a) 2 hours
b) 4 hours
c) 5 hours
d) 6 hours

Explain what the negative x-intercept represents in the given situation.
a) It represents the time when the balloon reaches its maximum altitude.
b) It represents the time when the balloon touches the ground.
c) It represents the time when the balloon reaches its initial starting point.
d) It represents the time when the balloon reaches its maximum speed.

Answer 1

1. The positive x-intercept of the function h(x) is 5 hours (option c).
2. The positive x-intercept of the function h(x) represents the time when the balloon touches the ground (option b).
3. the negative x-intercept to be approximately 5 hours (option c).
4. The negative x-intercept of the function h(x) represents the time when the balloon touches the ground. (option b).

Step-by-step explanation:

1) To find the positive x-intercept of the function h(x), we need to determine the value of x when h(x) equals zero.

The function h(x) is given by:

h(x) = -16x² + 64x + 80

To find the x-intercept, we set h(x) equal to zero:

0 = -16x^2 + 64x + 80

Next, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

For our equation, a = -16, b = 64, and c = 80. Substituting these values into the formula, we get:

x = (-64 ± √(64² - 4(-16)(80))) / (2(-16))

Simplifying the equation further, we have:

x = (-64 ± √(4096 + 5120)) / (-32)

x = (-64 ± √9216) / (-32)

x = (-64 ± 96) / (-32)

Now we have two possible solutions:

x = (-64 + 96) / (-32) = 32 / (-32) = -1

x = (-64 - 96) / (-32) = -160 / (-32) = 5

Since we are looking for the positive x-intercept, we take the value of x that is positive, which is 5.

Therefore, the positive x-intercept of the function h(x) is 5 hours (option c).

So, the correct answer is c

2) The x-intercept of a function occurs when the value of the dependent variable, in this case, the altitude (h), is equal to zero. In the context of the given situation, when the balloon touches the ground, its altitude will be zero.

So, when we solve the quadratic equation -16x² + 64x + 80 = 0, we find the value of x that corresponds to the time when the balloon is on the ground. This value is the positive x-intercept.

In this case, the positive x-intercept is 5 hours, which means that 5 hours after starting its ascent, the balloon touches the ground.

It's important to note that the x-intercept does not represent the time when the balloon reaches its maximum altitude (option a), reaches its initial starting point (option c), or reaches its maximum speed (option d). Those concepts are not directly related to the x-intercept in this context.

Therefore, the correct answer is b

3) The negative x-intercept of the function h(x) can be found by setting h(x) equal to zero and solving for x:

-16x² + 64x + 80 = 0

To solve this quadratic equation, we can factor it or use the quadratic formula. In this case, factoring is not feasible, so we can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

For our equation, a = -16, b = 64, and c = 80. Plugging these values into the quadratic formula, we get:

x = (-64 ± √(64² - 4(-16)(80))) / (2(-16))

After simplifying, we find the negative x-intercept to be approximately 5 hours (option c).

Therefore, the correct answer is c

4) In the given situation, the function h(x) represents the altitude of the balloon x hours after it starts its ascent from the hill. The x-intercepts of a function are the points where the graph of the function intersects the x-axis. Since the altitude of the balloon is represented by the function h(x), the x-intercepts of the function correspond to the points where the altitude of the balloon is zero.

When the altitude of the balloon is zero, it means the balloon has touched the ground. Therefore, the negative x-intercept of the function h(x) represents the time when the balloon touches the ground.

Therefore, the correct answer is b) It represents the time when the balloon touches the ground.