Power function story problem!

**wiseguy** · July 24th 2010, 08:17 PM

I have several problems like this one in my book. It's asking me to find out an equation using a table of values.

"A precalculus class launches a model rocket on the football field. the rocket fires for two seconds. Each second thereafter they measure its altitude, finding these values.
t(s) h(ft)
2 166
3 216
4 234
5 220
6 174

At what time t do you predict the rocket will hit the ground? (Round to the nearest hundredth)"

I already know this is a power function, because the rocket will eventually hit the ground, something the exponential doesn't do. I just don't see how to use 234 feet, 220 feet, and 174 feet.

I'm confused out of my mind, and if I could get some guidance on this one problem I'll certainty be able to figure out the slew of other ones like this!

Thanks in advance ya'll

**TKHunny** · July 24th 2010, 08:48 PM

What did you plan to do with the 166 and the 216?

What tools do you get? Can you use the five points to find the maximum? It's unlikely to happen at a measurement point. I used a method that suggests 4.12 sec. What do you get?

Knowing 4.12 sec MAY suggest a landing time of 8.24 sec, but that might be ignoring the first two seconds.

Another method suggests a landing time of 8.103 seconds.

You must figure out how to use your data.

**wiseguy** · July 25th 2010, 07:42 AM

That's the issue, I'm at a total loss on how to use the data. Any guidance would be great since the book is very shady on this and I have other problems that I could use this information with to figure them out

**skeeter** · July 25th 2010, 08:25 AM

Originally Posted by wiseguy

That's the issue, I'm at a total loss on how to use the data. Any guidance would be great since the book is very shady on this and I have other problems that I could use this information with to figure them out

plot the data in a graphing calculator and perform a quadratic regression ...

**wiseguy** · July 25th 2010, 08:28 AM

Nice!

Is this the only approach, as in I cannot "show work" for this sort of problem?

**wiseguy** · July 25th 2010, 08:31 AM

Here's what my calculator gave me:

$-16x^2+130x-30$

Can I write this out and show work?

**skeeter** · July 25th 2010, 09:05 AM

Originally Posted by wiseguy

Here's what my calculator gave me:

$-16x^2+130x-30$

Can I write this out and show work?

first off, remember that the independent variable is t, not x ... and the domain of t is from t = 2 until t = whenever the rocket hits the ground.

one method to do this by hand is to use at least three of the points to get three equations ...

$a(2^2) + b(2) + c = 166$

$a(3^2) + b(3) + c = 216$

$a(6^2) + b(6) + c = 174<br />$

now, solve the system for the coefficients a, b, and c.

**wiseguy** · July 25th 2010, 09:16 AM

Would using the first two work? The question asks (another similar one) to use just two:

The rate at which water flows out of a hose is function of the water pressure at the faucet. Suppose that these flow rates have been measured (psi is "pounds per square inch").

x(psi) y(gal/min)
4 5.0
9 7.5
16 10.0
25 12.5
36 15.0

Based on physical considerations, the flow rate is expected to be a power function of the pressure. Use the first and second data points to find the particular equation of the power function.

**skeeter** · July 25th 2010, 09:24 AM

what is the form of a power function?

**wiseguy** · July 25th 2010, 09:29 AM

I know what the form of a power function is, I've been spending quite some time on this problem. No where in the book does it show how a square in x values can relate to linear y values.

To answer your question, f(x)=ax^b

The rate at which water flows out of a hose is function of the water pressure at the faucet. Suppose that these flow rates have been measured (psi is "pounds per square inch").

x(psi) y(gal/min)
4 5.0
9 7.5
16 10.0
25 12.5
36 15.0

Based on physical considerations, the flow rate is expected to be a power function of the pressure. Use the first and second data points to find the particular equation of the power function.

**skeeter** · July 25th 2010, 09:40 AM

$y = ax^b$

$5 = a \cdot 4^b$

$7.5 = a \cdot 9^b$

solve the system for a and b. this should get you started ...

$\displaystyle \frac{7.5}{5} = \frac{a \cdot 9^b}{a \cdot 4^b}$

**wiseguy** · July 25th 2010, 09:49 AM

The textbook says:

"The rate at which water flows out of a hose is function of the water pressure at the faucet. Suppose that these flow rates have been measured (psi is "pounds per square inch").

x(psi) y(gal/min)
4 5.0
9 7.5
16 10.0
25 12.5
36 15.0

Based on physical considerations, the flow rate is expected to be a power function of the pressure. Use the first and second data points to find the particular equation of the power function."

So I took their method, but what comes out is garbage:

a*2^2=5
a*3^2=7.5

**skeeter** · July 25th 2010, 09:56 AM

one more time ...

$y = ax^b$

$5 = a \cdot 4^b$

$7.5 = a \cdot 9^b$

solve the system for a and b. this should get you started ...

$\displaystyle \frac{7.5}{5} = \frac{a \cdot 9^b}{a \cdot 4^b}$

**wiseguy** · July 25th 2010, 09:59 AM

Since there are two variables, won't it be impossible to find them both?

**skeeter** · July 25th 2010, 10:05 AM

what happens to the value of "a" when you divide the equations as set up?

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