Height of Rocket Over Sloping Ground - PRECAL INVERSE FUNCTION PROBLEM

In summary: Now, you do not know the value of $f$, but you can find it by using your value of $f_{\max}=1922$ and solve for $f$ at $h=1922$:1922=-2x^2+124xUsing the quadratic formula to solve for $x$ yields:x=\frac{124\pm\sqrt{124^2+4(2)(1922)}}{2(2)}=\frac{124\pm2\sqrt{3844+2(1922)}}{4}=\frac{124\pm2\sqrt{7690}}{4}The value of $x$ at $h=1922$ is the larger of the
  • #1
MarkFL
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Here is the question:

PRECAL INVERSE FUNCTION PROBLEM PLEASE HELP?


Clovis is standing at the edge of a cliff, which slopes 4 feet downward from him for every 1 horizontal foot. He launches a small model rocket from where he is standing.

With the origin of the coordinate system located where he is standing, and the x-axis extending horizontally, the path of the rocket is described by the formula y= -2x^2 + 120x.

a.) Give a function h = f(x) relating the height h of the rocket above the sloping ground to its x-coordinate.

0 = -2x^2 +124

b.) Find the maximum height of the rocket above the sloping ground. What is its x-coordinate when it is at its maximum height?

MAX: 31, 1922

c.) Clovis measures its height h of the rocket above the sloping ground while it is going up. Give a function x = g(h) relating the x-coordinate of the rocket to h.

The maximum I get for the rocket is (31, 1922)
I am pretty sure it is right because the answer in the back of the book is 31 - 1/2 sqrt3844 - 2h
so the 31 is probably correct!
I just need the answer to C (obviously), I'd really appreciate it if you could guide me through it rather than just answers.
Please no use of Calculus terms, I am in Precalc.

I have posted a link there to this thread so the OP can view my work.
 
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  • #2
Hello Robin,

We are given the trajectory of the rocket as:

\(\displaystyle h(x)=-2x^2+120x\)

And we may model the surface of the ground by observing it passes through the origin of our coordinate system and has a slope of $-4$. Thus the "ground" function is:

\(\displaystyle g(x)=-4x\)

a.) Give a function h = f(x) relating the height h of the rocket above the sloping ground to its x-coordinate.

The height of the rocket above the ground at $x$ is the difference between the two functions:

\(\displaystyle f(x)=h(x)-g(x)=\left(-2x^2+120x \right)-(-4x)=-2x^2+124x=2x(62-x)\)

I think you meant to give this but simply made a typo and left off the variable of the linear term.

b.) Find the maximum height of the rocket above the sloping ground. What is its x-coordinate when it is at its maximum height?

From part a) we see the two roots of $f(x)$ are:

\(\displaystyle x=0,\,62\)

Now, we know the axis of symmetry, along which the vertex lies, will be midway between the roots. Using the mid-point formula, we then find the axis of symmetry is:

\(\displaystyle x=\frac{0+62}{2}=31\)

The height at the value of $x$ is:

\(\displaystyle f(31)=2\cdot31(62-31)=2\cdot31^2=1922\)

Hence:

\(\displaystyle f_{\max}=f(31)=1922\)

We have found that the rocket reaches a maximum height of 1922 ft at a horizontal distance of 31 ft from the origin.

This agrees with what you found. (Clapping)

c.) Clovis measures its height h of the rocket above the sloping ground while it is going up. Give a function x = g(h) relating the x-coordinate of the rocket to h.

Recall we found:

\(\displaystyle f(x)=-2x^2+124x\)

Arrange in standard quadratic form:

\(\displaystyle 2x^2-124x+f=0\)

Application of the quadratic formula yields:

\(\displaystyle x=\frac{124\pm\sqrt{(-124)^2-4(2)(f)}}{2(2)}=\frac{62\pm\sqrt{3844-2f}}{2}\)

Now, since $f$ is not a one-to-one function, we must observe that the relevant domain of $f$ is \(\displaystyle 0\le x\le62\) and so we must take the branch:

\(\displaystyle x=\frac{62-\sqrt{3844-2f}}{2}=31-\frac{\sqrt{3844-2f}}{2}\)
 

FAQ: Height of Rocket Over Sloping Ground - PRECAL INVERSE FUNCTION PROBLEM

What is the purpose of studying the height of a rocket over sloping ground?

The purpose of studying the height of a rocket over sloping ground is to understand the trajectory and flight path of a rocket launch. This information can help scientists and engineers design and optimize rocket launches for various purposes such as space exploration, satellite launches, and military operations.

What is the main concept behind the precal inverse function problem related to the height of a rocket over sloping ground?

The main concept behind the precal inverse function problem is to determine the initial angle at which the rocket must be launched in order to reach a specific height over a sloping ground. This involves using inverse trigonometric functions to solve for the initial angle, given the distance traveled, height reached, and slope of the ground.

What factors affect the height of a rocket over sloping ground?

The height of a rocket over sloping ground is affected by several factors including the initial velocity of the rocket, the angle at which it is launched, the slope of the ground, air resistance, and the force of gravity. These factors must be considered in order to accurately predict the height of a rocket at any given point during its flight.

How is the inverse function problem related to rocket launches used in real-world applications?

The inverse function problem related to rocket launches is used in real-world applications such as designing and optimizing rocket launches for space exploration, satellite launches, and military operations. It is also used in the development of mathematical models for predicting the trajectory and flight path of rockets, as well as in the analysis of data from actual rocket launches.

What are some challenges in solving the precal inverse function problem for the height of a rocket over sloping ground?

Some challenges in solving the precal inverse function problem for the height of a rocket over sloping ground include accurately measuring the initial velocity and angle of the rocket, accounting for variations in air resistance and gravity, and taking into consideration the curvature of the Earth. Additionally, real-world factors such as wind, weather conditions, and other external forces can also affect the height of a rocket over sloping ground, making it a complex and challenging problem to solve.

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