Question 661979
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=(54-35)/(12-5)}}} --> {{{m=19/7}}}.
Then we can use that slope and the coordinates of one of the points to write the equation in point-slope form.
{{{y-35=(19/7)(x-5)}}}
From the point-slope form, we can go to the slope-intercept form, which will help answer the first 2 questions:
{{{y-35=(19/7)(x-5)}}}-->{{{y-35=(19/7)x-5(19/7)}}}-->{{{y-35=(19/7)x-95/7}}}-->{{{y=(19/7)x-95/7+35}}}-->{{{y=(19/7)x-95/7+35}}}-->{{{y=(19/7)x+150/7}}}
With a calculator, we can use the approximation {{{y=2.714x+21.43}}}
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=(19/7)*18+150/7}}}-->{{{y=342/7+150/7}}}-->{{{y=492/7=70&2/7}}}=about{{{70.29}}}=about{{{70}}} (rounding)
I assume the expected answer is {{{70pounds}}} rounding to whole numbers.
For the 20 foot tree,
{{{x=20}}} --> {{{y=(19/7)*20+150/7}}}-->{{{y=380/7+150/7}}}-->{{{y=530/7=75&5/7}}}=about{{{75.71}}}=about{{{76}}} (rounding)
I assume the expected answer is {{{76pounds}}} rounding to whole numbers.
For the 88 pound production, {{{y=88}}}, so
{{{88=(19/7)x+150/7}}}-->{{{88-150/7=(19/7)x}}}-->{{{616/7-150/7=(19/7)x}}}-->{{{466/7=(19/7)x}}}
Multiplying both sides times {{{7}}}:
{{{466/7=(19/7)x}}}-->{{{466=19x}}}-->{{{x=466/19=24&10/19}}}=about{{{24.53}}}=about{{{25}}} (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*x+b}}}
by substituting the coordinates for out 2 points and solving the system
{{{system(54=12m+b,35=5m+b)}}}
Solving the system of equations leads to
{{{m=19/7}}}=about{{{2.714}}} (rounding)
and {{{b=150/7}}}=about{{{21.43}}} (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 {{{492/7}}} or {{{70&2/7}}} pounds of oranges for an 18 foot tree.
=FORECAST(20,B1:B2,A1:A2) gives me {{{75.71}}} or {{{530/7}}} or {{{75&5/7}}} 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.