Question

1. (l4) jessica and amanda are growing sunflowers in their backyard. jessica’s sunflower starts at a height of 8 inches, and grows 2 inches each week. amanda’s sunflower starts at a height of 5 inches, and grows 20% each week. in about how many weeks will amanda’s sunflower be taller than jessica’s sunflower? (hint: use desmos)

Explanation:

Step1: Set up height - functions

Let $w$ be the number of weeks. Jessica's sun - flower height function is $J(w)=8 + 2w$. Amanda's sun - flower height function is $A(w)=5+(5\times0.2)w=5 + w$.

Step2: Set up the inequality

We want to find $w$ such that $A(w)>J(w)$. So, $5 + w>8 + 2w$.

Step3: Solve the inequality

Subtract $w$ from both sides: $5>8 + 2w-w$, which simplifies to $5>8 + w$. Then subtract 8 from both sides: $5 - 8>w$, or $w < - 3$. This is incorrect. Let's correct the functions. Amanda's sun - flower grows 20% of its initial height each week. The correct function for Amanda's sun - flower height is $A(w)=5\times(1 + 0.2)^w$.
We want $5\times(1.2)^w>8 + 2w$.
We can use a graphing utility (Desmos as suggested). Graph $y = 5\times(1.2)^w$ and $y=8 + 2w$.
By looking at the intersection point on Desmos, we find the value of $w$ for which the exponential function $y = 5\times(1.2)^w$ is above the linear function $y = 8+2w$.
When we graph these two functions on Desmos, we find that $w\approx7$ weeks.

Answer:

7 weeks