Exercise 119 Page 143

Some time ago, Josie moved into a new house. She recorded the following data about the growth of a beautiful vine that grows on her back fence. Assuming that the vine continued to grow at a constant rate, we want to find the height of the vine 7 weeks after Josie moved into her house.

Weeks Since Josie Moved In	4	5	6
Height of Vine (in inches)	16	19	22

First, let's focus on the height of the vine in weeks 5 and 6. If we divide the difference in heights by the time that passed, we will get the rate of growth of the vine. Difference in Height/Time Passed After 5 weeks the vine was 19 inches tall. After 6 weeks the vine was 22 inches tall. Let's substitute the values into our expression. 22 in.- 19in./6 weeks - 5 weeks Simplify the quotient.

22 in.-19in./6 weeks - 5 weeks

SubTerms

Subtract terms

3in./1 week

We found that the vine grew 3 inches per week. We are told that the rate of growth is constant. This means that 7 weeks after Josie moved into the house, the vine was 3 inches taller than the week earlier. Let's now calculate the height of the vine after 7 weeks. Week $7$: 22 + 3 = 25 in. The vine was 25 inches tall 7 weeks after Josie moved in.

This time we want to find how tall the vine was 3 weeks after Josie moved in. Let's take another look at the table.

Weeks Since Josie Moved In	4	5	6
Height of Vine (in inches)	16	19	22

In Part A we found the weekly rate of growth of the vine. 3in./1 week We want to find the height at 3 weeks. Notice that we only know the height after 4 weeks. This means we need to reverse the pattern. We are going to subtract the weekly growth from the height after 4 weeks. Let's do it! Week $3$: 16 -3 = 13 in. The vine was 13 inches tall 3 weeks after Josie moved in.

Now we want to find how tall the vine was when Josie moved in. Here is the table monitoring the growth of the plant.

Weeks Since Josie Moved In	4	5	6
Height of Vine (in inches)	16	19	22

We know that 4 weeks after Josie moved into the house, the vine was 16 inches tall. We also know that the vine was growing at a constant rate of 3 inches per week. We can use this fact to calculate how much the vine grew through the first 4 weeks. 4(3 in.) = 12 in. Now let's subtract the value from the height after 4 weeks. This will give us the initial height of the vine. Week $0$: 16 -12 = 4 in. When Josie moved in, the vine was 4 inches tall.

We want to predict how tall the vine was 19 weeks after Josie moved in. To do so, let's summarize what we have found so far. Week $0$:& 4 in. Rate of Growth:& 3 in. per week

We can see that the vine was 4 inches tall when Josie moved into her house, and the then vine grew 3 inches every week. Let's now calculate how much the vine grew through 19 weeks. 19(3 in.) = 57 in. In 19 weeks the vine grew 57 inches. If we add this to the initial height of the plant, we will get the total height after 19 weeks. Week $19$: 4+ 57 = 61 in. The vine was 61 inches tall 19 weeks after Josie moved in.

Finally, we want to predict when the vine reached the top of the 94-inch-tall garage. Before we do that, let's summarize the relevant information we have gathered so far. Week $0$:& 4 in. Rate of Growth:& 3 in. per week We will use the information above to write a formula for the height of the vine after any number of weeks. Let n be that number. In order to find the height after some time, we need to multiply the number of weeks by the weekly growth rate. This will give us the total growth over that period of time. 3n Now, we also need to include the initial height of the vine. Let's do it! 3n + 4 This is our formula for the height of the vine after n weeks. Now, we want to find how many weeks it takes for the vine to reach a height of 94 inches. Let's set our expression equal to 94. 3n+4=94 We can now solve the equation for n.

3n+4=94

SubEqn

LHS-4=RHS-4

3n=90

DivEqn

.LHS /3.=.RHS /3.

3n/3=90/3

SimpQuot

Simplify quotient

n=90/3

CalcQuot

Calculate quotient