Question

1. joey was trying to see how many flowers could fit into his vases of various heights. he wrote down the information from the vases he currently had. he will receive a shipment of 4 more vases. the heights of the vases will be 14 in, 22 in, 2 feet, and 2.5 feet. x represents the height of a vase in inches and y represents the number of flowers that can fit in the vase.

x	y
8	5
12	7
18	10

how many flowers will fit in each new vase?

complete this sentence: for every __ inch(es) increase in the height of the vase, __ more flower(s) can fit into the vase.

Explanation:

Step1: Find the rate of change

First, find the change in \(y\) (number of flowers) and change in \(x\) (height of vase) between two - points. Let's take the first two points \((x_1,y_1)=(6,4)\) and \((x_2,y_2)=(8,5)\). The change in \(x\) is \(\Delta x=x_2 - x_1=8 - 6 = 2\), and the change in \(y\) is \(\Delta y=y_2 - y_1=5 - 4 = 1\). The rate of change (slope \(m\)) is \(\frac{\Delta y}{\Delta x}=\frac{1}{2}\). We can check with other points. For \((x_1,y_1)=(8,5)\) and \((x_2,y_2)=(12,7)\), \(\Delta x = 12-8 = 4\) and \(\Delta y=7 - 5 = 2\), and \(\frac{\Delta y}{\Delta x}=\frac{2}{4}=\frac{1}{2}\). So for every 2 - inch increase in the height of the vase, 1 more flower can fit into the vase.

Step2: Convert the new vase heights to inches

The new vase heights are 14 in, 22 in, 2 feet, and 2.5 feet. Since 1 foot = 12 inches, 2 feet \(=2\times12 = 24\) inches and 2.5 feet \(=2.5\times12=30\) inches.

Step3: Use the linear relationship \(y - y_1=m(x - x_1)\)

We use the point - slope form of a line \(y - y_1=m(x - x_1)\) with \(m=\frac{1}{2}\). Let's use the point \((x_1,y_1)=(6,4)\).
For \(x = 14\):
\[y-4=\frac{1}{2}(14 - 6)\]
\[y-4=\frac{1}{2}\times8\]
\[y-4 = 4\]
\[y=8\]
For \(x = 22\):
\[y-4=\frac{1}{2}(22 - 6)\]
\[y-4=\frac{1}{2}\times16\]
\[y-4 = 8\]
\[y = 12\]
For \(x = 24\):
\[y-4=\frac{1}{2}(24 - 6)\]
\[y-4=\frac{1}{2}\times18\]
\[y-4 = 9\]
\[y = 13\]
For \(x = 30\):
\[y-4=\frac{1}{2}(30 - 6)\]
\[y-4=\frac{1}{2}\times24\]
\[y-4 = 12\]
\[y = 16\]

Answer:

For vases of heights 14 in, 22 in, 24 in, 30 in, the number of flowers are 8, 12, 13, 16 respectively. For every 2 inch(es) increase in the height of the vase, 1 more flower(s) can fit into the vase.