Question

NO LINKS!!! URGENT HELP PLEASE!!! NOT MULTIPLE CHOICE!!!!

1. You and a friend take a hot air balloon ride for Valentine's Day. The path of the balloon can be modeled by the equation b(h) = 2h - 1/115h^2 in feet per minute. Use this scenario to answer questions a - c.

a. How is the balloon ride?

b. What is the maximum height the balloon reaches?

c. When you have been on the ride for 180 minutes, what is the height of the balloon? (round your answer to the nearest foot)

Answer 1

a) The balloon ride is 230 minutes long.

b) The maximum height the balloon reaches is 115 m.

c) The height of the balloon at 180 minutes is 78 feet.

Explanation:

Given quadratic equation:

`b(h)=2h-(1)/(115)h^2`

Part a

The height of the balloon at the start and end of the balloon ride is zero feet. Therefore, to determine how long the balloon ride is, set the given quadratic equation to zero and solve for h.

`\begin{aligned}\implies2h-(1)/(115)h^2&=0\\\\h\left(2-(1)/(115)h\right)&=0\\\\2-(1)/(115)h&=0\\\\(1)/(115)h&=2\\\\h&=230\end{aligned}`

Therefore, the balloon ride is 230 minutes long.

Part b

To determine the maximum height the balloon reaches, find the y-value of the vertex of the given quadratic equation.

The formula for the x-value of the vertex is -b/2a for a quadratic equation in the form y=ax²+bx+c. Therefore, the x-value of the vertex of the given equation is:

`\implies -(2)/(2\left(-(1)/(115)\right))=115`

To find the y-value of the vertex, substitute h = 115 into the given equation:

`\begin{aligned}\implies b(115)&=2(115)-(1)/(115)(115)^2\\&=230-115\\&=115\; \sf ft\end{aligned}`

Therefore, the maximum height the balloon reaches is 115 ft.

Part c

To determine the height of the balloon when you have been on the ride for 180 minutes, substitute h = 180 into the equation:

`\begin{aligned}\implies b(180)&=2(180)-(1)/(115)(180)^2\\\\&=360-(6480)/(23)\\\\&=78.260869...\\\\&=78\; \sf ft\end{aligned}`

Therefore, the height of the balloon at 180 minutes into the ride is 78 feet (rounded to the nearest foot).

Answer 2

1.

a.

b. 115 feet.

c. 78.26feet.

Explanation:

a.

some mathematical observations we can make about the path of the balloon based on the given equation:

• The balloon's vertical velocity decreases as the balloon rises. This is because the second term in the equation, 1/115h^2, becomes larger and larger as h decreases, which causes the velocity to decrease.
• The balloon's vertical velocity is zero at two points: h = 0 and h = 230. At h = 0, the balloon is on the ground and has not yet started to rise, so its velocity is zero. At h = 230, the balloon has reached its maximum height and has stopped rising, so its velocity is also zero.
• The balloon's vertical velocity is positive when h is less than 115, and negative when h is greater than 115. This means that the balloon is rising when its height is less than 115, and descending when its height is greater than 115.
• The maximum height the balloon can reach is 115 feet, which occurs at h = 115. At this height, the balloon's velocity is 1.74 feet per minute.
• The balloon cannot fly below a certain height, which is the vertical asymptote at h = 0. This means that the balloon cannot go below the ground level.

b. The maximum height of the balloon occurs at the vertex of the parabola described by the function b(h). We can find the height of the vertex by using the formula h = -b/(2a), where b and a are the coefficients of the linear and quadratic terms in the equation, respectively.

In this case, a = -1/115 and b = 2, so the height of the vertex is:

h = -b/(2a) = -2/(2(-1/115)) = 115 feet

Therefore, the maximum height the balloon reaches is 115 feet.

c. To find the height of the balloon after 180 minutes, we can substitute h = 180 into the equation b(h) and simplify:

b(180) = 2(180) - 1/115(180)^2 = 360 - 281.73≈ 78.26

Therefore, when you have been on the ride for 180 minutes, the height of the balloon is approximately78.26feet.