word problem help

Question

The percentage of soldiers who are unavailable for combat has increased approximately linearly from 11% in 2007 to 15% in 2010. The main causes are due to repeated deployments and health problems. Let p be the percentage of soldiers unavailable for combat at t years since 2000. Find an equation of a linear model to describe the data.

Answer 1

p represents percentage on the y axis, and t represents time on the x axis.

The sentence, Let p be the percentage of soldiers unavailable for combat at t years since 2000, describes the origin, where x and y are equal to zero.

They give two points (7, 11) and (10, 15), Solve for the slope and get 4/3.

p = mt + b, m=4/3.  Pick one of the two points, say (7,11).

so 11 = (4/3) * 7 + b

Solve for b.  Plug m and b back into p = mt + b and there is your answer.

Can you take it from there Tyler?

(If you wanted to take the other point (10, 15) and solve for b, you should get the same b as above, it is your check if you like)

Answer 2

Set 2000 yr as  t0 = 0

 

therefore, 2007 yr is t1` = t0 + 7 = 7 yr

        and  2010 yr is t2 = t0 + 10 = 10 yr

   

Let p1 = 11% of  soldiers unavailable for combat during 2007

 

      p2 = 15% of soldiers unavailable for combat during 2010

 

      m = slope of line

 

      b = y-intercept

 

1)  p1 = m*t1 + b

 

2)  p2 = m*t2 + b

-----------------------

 

Subtract equation 2) from equation 1) and solve for slope, m:

 

(p1 - p2) = m*(t1 - t2)

m = (p1 - p2)/(t1 - t2) = (11 - 15)/(7 - 10) = -4/-3 = 4/3 

 

m = 3/4 % of soldiers unavailable/yr

 

Plug m into equation 1) (or equation 2) and solve for y-intercept, b

 

p1 = m*t1 + b

 

b = p1 - m*t1 = 11 - (3/4)*7 = 11 - 21/4 = 44/4 - 21/4 = 23/4 or 5 3/4

 

b = 5 3/4 %

 

Now plug in standard equation of line with slope, m, and the y-intercept, b:

 

p = (3/4)*t + 5 3/4

 

where: p = % of soldiers unavailable for combat

           t = number of years since 2000