What could the number 3.5 represent in the equation?

Question

At the beginning of spring, Dominic planted a small sunflower in his backyard. The sunflower's height in inches, h, after w weeks, is given by the equation h=19+3.5w. What could the number 3.5 represent in the equation?

A. The sunflower's final height
B. The sunflower's height when it was planted.
C. The sunflower's height after one week.
D. The change in the sunflower's height for every one additional week.

The question above is confusing to me since to me it has 2 potential answers C. and D. I can't figure out if these two answers have separate meanings or not. Please explain.

Answer 1

The answer C means the complete height of the sunflower at the end of the first week after planing it.

The answer D means one measures on a day, waits a week, measures again and records the change, the difference between the two measurements.

Note that when the sunflower was planted it already had some height. This height also appears in the equation.

Answer 2

After one week, the sunflower height is $h(1) = 22.5$, so the answer cannot be C. Now, suppose $h(w_{0}) = 19+3,5w_{0}$ is the height of the sunflower at some week $w_{0}$. After one week, its height become: $$h(w_{0}+1) = 19+3,5(w_{0}+1) = 19+3,5w_{0} + 3,5 = h(w_{0}) + 3,5$$ Thus, the change in the sunflower's height after one week is 3,5 inches. Since $w_{0}$ was assumed to be an arbitrary week, the sunflower will always grow $3,5$ inches after one week.

Answer 3

Replacing $h$ with $y$ and $w $ with $x$, we have $y=19+3.5x=3.5x+19$. This is a standard equation for straight line $y=mx+b $, where slope m= 3.5 and $y$ intercept b=19.

The original height is $y(0)=3.5 \times 0+ 19=19=b$.

After 1 week , the height is $y(1)=3.5\times 1+19=22.5$

Similarly, after n weeks, the height is $y(n)=3.5\times n+19$, after $n+1$ weeks, the height is $y(n+1)= 3.5\times (n+1)+19$, the change for every additional week is $y(n+1)-y(n)=3.5=m$.

Therefore, the correct answer is (D).