SOLUTION: The yearly production of a 5 foot orange tree is 35 pounds of oranges. A 12 foot tree produces 54 pounds. What is the predicted production of an 18 foot tree? What is the predic

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: The yearly production of a 5 foot orange tree is 35 pounds of oranges. A 12 foot tree produces 54 pounds. What is the predicted production of an 18 foot tree? What is the predic      Log On


   

Question 661979: The yearly production of a 5 foot orange tree is 35 pounds of oranges. A 12 foot tree produces 54 pounds.
What is the predicted production of an 18 foot tree?
What is the predicted production of an 20 foot tree?
What is the predicted height ( in feet ) of a tree that produces 88 pounds of oranges?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
If the production, y of an orange tree varies linearly with its height, x,
we can write the linear equation that describes it, knowing two (x,y) points.
"The yearly production of a 5 foot orange tree is 35 pounds of oranges," gives us point (5,35).
"A 12 foot tree produces 54 pounds," gives us point (12,54).

I can see 3 ways to get the answer.
The expected way to solve the problem depends on what you are studying in class.

CALCULATING SLOPE OF A LINE AND WRITING THE POINT-SLOPE FORM:
We can calculate the slope, m of the line:
m=%2854-35%29%2F%2812-5%29 --> m=19%2F7.
Then we can use that slope and the coordinates of one of the points to write the equation in point-slope form.
y-35=%2819%2F7%29%28x-5%29
From the point-slope form, we can go to the slope-intercept form, which will help answer the first 2 questions:
y-35=%2819%2F7%29%28x-5%29-->y-35=%2819%2F7%29x-5%2819%2F7%29-->y-35=%2819%2F7%29x-95%2F7-->y=%2819%2F7%29x-95%2F7%2B35-->y=%2819%2F7%29x-95%2F7%2B35-->y=%2819%2F7%29x%2B150%2F7
With a calculator, we can use the approximation y=2.714x%2B21.43
and get accurate enough results.
I'll stick with the fractions just in case you are expected to do that.
For the 18 foot tree, x=18, so
y=%2819%2F7%29%2A18%2B150%2F7-->y=342%2F7%2B150%2F7-->y=492%2F7=70%262%2F7=about70.29=about70 (rounding)
I assume the expected answer is 70pounds rounding to whole numbers.
For the 20 foot tree,
x=20 --> y=%2819%2F7%29%2A20%2B150%2F7-->y=380%2F7%2B150%2F7-->y=530%2F7=75%265%2F7=about75.71=about76 (rounding)
I assume the expected answer is 76pounds rounding to whole numbers.
For the 88 pound production, y=88, so
88=%2819%2F7%29x%2B150%2F7-->88-150%2F7=%2819%2F7%29x-->616%2F7-150%2F7=%2819%2F7%29x-->466%2F7=%2819%2F7%29x
Multiplying both sides times 7:
466%2F7=%2819%2F7%29x-->466=19x-->x=466%2F19=24%2610%2F19=about24.53=about25 (rounding)
I assume the expected answer is 25feet rounding to whole numbers.

SETTING UP A SYSTEM OF EQUATIONS TO FIND SLOPE AND INTERCEPT:
WE can find an equation of the form y=m%2Ax%2Bb
by substituting the coordinates for out 2 points and solving the system
system%2854=12m%2Bb%2C35=5m%2Bb%29
Solving the system of equations leads to
m=19%2F7=about2.714 (rounding)
and b=150%2F7=about21.43 (rounding)
From that point on, the calculations are the same as done above.

USING A COMPUTER/CALCULATOR AS IF IT IS A STATISTICS PROBLEM
Using Excel functions in my computer, I can answer all 3 questions.
With A1=5, A2=12, B1=35, B2=54,
=FORECAST(18,B1:B2,A1:A2) gives me 70.29 or 492%2F7 or 70%262%2F7 pounds of oranges for an 18 foot tree.
=FORECAST(20,B1:B2,A1:A2) gives me 75.71 or 530%2F7 or 75%265%2F7 pounds of oranges for an 20 foot tree.
=FORECAST(88,A1:A2,B1:B2) gives me 24.53 feet for the height of a tree that would produce 88 pounds of oranges in a year.