Question

marcus is trying to lose weight for next football season. four weeks into his diet, he weighed 232 pounds. at nine weeks, he weighed 224.5 pounds.
a) write an equation to model marcus weight, y, after x weeks into his diet.
b) what does the slope & y - intercept mean in terms of the problem?
c) how many weeks will it take marcus to reach his goal weight of 214 pounds?

Explanation:

Step1: Find the slope

The formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(4,232)\) and \((x_2,y_2)=(9,224.5)\).
\(m=\frac{224.5 - 232}{9 - 4}=\frac{-7.5}{5}=-1.5\)

Step2: Use the point - slope form \(y - y_1=m(x - x_1)\)

Using \(m=-1.5\) and \((x_1,y_1)=(4,232)\)
\(y - 232=-1.5(x - 4)\)
\(y-232=-1.5x + 6\)
\(y=-1.5x+238\)

Step3: Interpret the slope and y - intercept

The slope \(m = - 1.5\) means Marcus loses \(1.5\) pounds per week. The y - intercept \(b = 238\) means Marcus' initial weight (at \(x = 0\) weeks) was \(238\) pounds.

Step4: Solve for \(x\) when \(y = 214\)

Substitute \(y = 214\) into \(y=-1.5x + 238\)
\(214=-1.5x+238\)
\(1.5x=238 - 214\)
\(1.5x=24\)
\(x = 16\)

Answer:

a) The equation is \(y=-1.5x + 238\).
b) The slope \(-1.5\) means Marcus loses \(1.5\) pounds per week. The y - intercept \(238\) means Marcus' initial weight was \(238\) pounds.
c) It will take \(16\) weeks.