Question

Based on this sequence, will Leonard ever give Penny 51 roses?If yes, on which day? If not, explain why not.

Answer 1

Leonard will give Penny 51 roses.

This will happen on the 11th day

Step-by-step explanation:

To detrmine if Leonard will ever give Penny 51 roses, we need to find the equation of the values we have in the table

Equation of line:

y = mx + b

m = slope, b = y-intercept

First let's find the slope of the line:

Using any two points on the table: (1, 1) and (2, 6)

`\begin{gathered} \text{Slope formula:} \\ m\text{ = }(y_2-y_1)/(x_2-x_1) \\ x_1=1,y_1=1x_2=2,y_2\text{ = }6 \\ m\text{ = }(6-1)/(2-1) \\ m\text{ = 5/1 = 5} \end{gathered}`

Slope of the line = 5

We need to find the y-intercept. Using the slope and any of the two points:

`\begin{gathered} y\text{ = mx + b} \\ point\text{ (1, 1): x = 1, y = 1} \\ 1\text{ = 5(1) + b} \\ 1\text{ = 5 + b} \\ 1-5\text{ = b} \\ b\text{ = -4} \end{gathered}`

The equation of the line:

`\begin{gathered} \text{y = 5x + (-4)} \\ y\text{ = 5x - 4} \end{gathered}`

To determine if Penny gets 51 roses, we will substitute 51 for y in our equation:

`\begin{gathered} 51\text{ = 5x - 4} \\ \text{Add 4 to both sides:} \\ 51\text{ + 4 = 5x - 4 + 4} \\ 55\text{ = 5x} \\ \\ \text{divide both sides by 5:} \\ (55)/(5)\text{ = }(5x)/(5) \\ x\text{ = 11} \end{gathered}`

From our calculaton, x = number of days and y = number of roses

When number of roses was 51, the day was on the 11th

Hence, Leonard will give Penny 51 roses.

This will happen on the 11th day