SOLUTION: Hello, First and foremost I would like to thank you for your assistance. I am a parent of a high school student and have not taken Alegbra in a years. I would appreciate any a

Question 39353: Hello,
First and foremost I would like to thank you for your assistance. I am a parent of a high school student and have not taken Alegbra in a years. I would appreciate any assistance you can provide.
Thanks,
Steve
Problem 1:

Use the table shown below. Assume the data can be represented as a linear relationship.

a) Using the two points defined by data from 2000 and 2005, develop a linear equation for the data.

b) Using the equation found in a), predict the tax revenue for 2006.

c) Use the calculator to determine the "best fit" equation for the data.

d) Using the equation found in c), predict the tax revenue for 2006.

e) Using the equation found in c), estimate tax revenue for the missing 2002 data

Year Sales Tax Revenue
2000 $19500
2001 $21000
2003 $24000
2004 $27300
2005 $31000

Note: data is missing for 2002, the county clerk embezzled the funds for that year.

__________________________________________________________________________

Problem #2

A rocket is shot from the top of an oceanside cliff that is 60ft high. The original angle is 45 degrees and the original velocity is 50ft per sec. The formula for the rocket's motion is:

y = f(x) = -0.0128x(squared) + x + 60

The variables x and y are both measured in feet. The variable x represents horizontal displacement (distance from the base of the cliff), and y represents the height of the rocket above the ground.

a) What is the maximum height attained by the rocket? How far has the rocket travelled horizontally at this point?

b) At waht point(s) (after launch) is the rocket 70 feet high? (Horizontal displacement)

c) How far away from the base of the cliff does the rocket land?
Found 2 solutions by stanbon, Nate:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
Problem 1:
Use the table shown below. Assume the data can be represented as a linear relationship.
a) Using the two points defined by data from 2000 and 2005, develop a linear equation for the data.
Points are (2000,19500) and (2005,31000)
Slope if [31000-19500/[2005-2000]=11500/5=2300
The form of the linear equation is y=mx+b where y=31000,x=2005,m=2300
31000=2300(2005)=b
b=-4580500
EQUATION: y=2300x-4580500
b) Using the equation found in a), predict the tax revenue for 2006.
tax=2300(2006)-4580500=$33,300
c) Use the calculator to determine the "best fit" equation for the data.
y=2204.65x-439074.42
d) Using the equation found in c), predict the tax revenue for 2006.
$32,056
e) Using the equation found in c), estimate tax revenue for the missing 2002 data
$23,237
Year Sales Tax Revenue
2000 $19500
2001 $21000
2003 $24000
2004 $27300
2005 $31000
Note: data is missing for 2002, the county clerk embezzled the funds for that year.

__________________________________________________________________________
Problem #2
A rocket is shot from the top of an oceanside cliff that is 60ft high. The original angle is 45 degrees and the original velocity is 50ft per sec. The formula for the rocket's motion is:
y = f(x) = -0.0128x(squared) + x + 60
The variables x and y are both measured in feet. The variable x represents horizontal displacement (distance from the base of the cliff), and y represents the height of the rocket above the ground.
a) What is the maximum height attained by the rocket? How far has the rocket travelled horizontally at this point?
The equation is a quadratic with a=-0.0128, b=1, c=60
The maximum point occurs at x=-b/2a=-1/2(-0.0128)=39.0625, height =79.53.. ft.
b) At waht point(s) (after launch) is the rocket 70 feet high? (Horizontal displacement)
80=-0.0128x^2+x+60
-0.0128x^2+x-10=0
x=11.774611 y=70; and x=66.350389 y=70

c) How far away from the base of the cliff does the rocket land?
0 solutions
Let -0.0128x^2+x+60=0
Solve for "x"
x=117.88749 ft
Cheers
Stan H.
Answer by Nate(3500) (Show Source): You can put this solution on YOUR website!
Year Sales Tax Revenue
2000 $19500
2001 $21000
2003 $24000
2004 $27300
2005 $31000
a) Using the two points defined by data from 2000 and 2005, develop a linear equation for the data.
2000 $19500 at this time (0 years from 2000)
2005 $31000 at this time (5 years from 2000)
y = ((31000-19500)/5)x + 19500
y = 2300x + 19500
b) Using the equation found in a), predict the tax revenue for 2006.
y = 2300(6) + 19500
y = 13800 + 19500 = 33300
c) Use the calculator to determine the "best fit" equation for the data.
I haven't done any equations dealing with the "best fit" line, but I will try my best. It seems right.
y = (31000-((19500 + 21000 + 24000 + 27300 + 31000)/5))x + 19500
y = 2205x + 19500
d) Using the equation found in c), predict the tax revenue for 2006.
y = 2205(6) + 19500 = 32730
e) Using the equation found in c), estimate tax revenue for the missing 2002 data
y = 2205x(2) + 19500 = 23910
.............................................................................

a) What is the maximum height attained by the rocket? How far has the rocket travelled horizontally at this point?
The highest point would be at the vertex.
((-b/2a),f(x))
10000/256 = 5000/128 = 2500/64 = 1250/32 = 625/16
f(x) = -0.0128(625/16)^2 + (625/16) + 60
v(39.0625,79.53125) Max height: 39.0625ft. Distance traveled: 79.53125ft.
b) At waht point(s) (after launch) is the rocket 70 feet high? (Horizontal displacement)

Use the quadratic formula to determine the root.

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=0.488 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: 11.7746106303547, 66.3503893696453. Here's your graph:

At about 11.8 ft. horizontally or about 66.4 ft.
c) How far away from the base of the cliff does the rocket land?
Again, use quadratic formula to determine the roots.

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=4.072 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: -39.762489097684, 117.887489097684. Here's your graph:

At about 117.9 ft. horizontally.

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