Question

brenna’s favorite book is about johnny appleseed, the american pioneer who planted apple trees all across the country. inspired by the story, brenna plants an apple seed in her backyard and tends the seed as it slowly grows into a tree. there is a proportional relationship between the age of brennas apple tree (in years), x, and the height of the tree (in feet), y.

Explanation:

Since the problem is about a proportional relationship between the age of the tree (x) and its height (y), we can assume we need to find the constant of proportionality or the equation of the line. Let's assume we need to find the equation of the proportional relationship (since the problem is about proportionality, the equation is \( y = kx \), where \( k \) is the constant of proportionality).

First, we need to find two points on the line. Let's look at the graph. From the grid, let's pick a point. For example, when \( x = 8 \) (assuming the x - axis is in years, and each grid square is 1 unit), what's the y - value? Wait, maybe a better point: let's see, when \( x = 8 \), maybe \( y = 8 \)? Wait, no, let's check the slope. Wait, the line passes through, let's say, when \( x = 8 \), \( y = 8 \)? Wait, no, let's take two clear points. Let's assume that when \( x = 8 \), \( y = 8 \)? Wait, no, maybe when \( x = 4 \), \( y = 4 \)? Wait, no, let's look at the graph again. Wait, the line starts from the origin (since it's a proportional relationship, so (0,0) is on the line) and then goes through, for example, (8, 8)? Wait, no, the y - axis is in feet. Wait, maybe when \( x = 8 \) years, \( y = 8 \) feet? Wait, no, let's calculate the slope. The slope \( k \) of a proportional relationship \( y=kx \) is \( k=\frac{y}{x} \). Let's take a point on the line. Let's say when \( x = 8 \), \( y = 8 \)? Wait, no, looking at the graph, the line goes through (8, 8)? Wait, no, the y - axis is labeled "Feet" and the x - axis is "Age (in years)". Let's take a point: when \( x = 8 \), \( y = 8 \)? Wait, maybe the line passes through (8, 8), so \( k=\frac{y}{x}=\frac{8}{8} = 1 \)? Wait, no, maybe I misread. Wait, let's take another point. Suppose when \( x = 4 \), \( y = 4 \)? No, maybe the line has a slope of 1. Wait, maybe the equation is \( y = x \), so the constant of proportionality \( k = 1 \). But maybe the problem is to find the equation of the line.

Wait, let's do it properly. For a proportional relationship \( y=kx \), we can find \( k \) by taking a point \( (x,y) \) on the line. Let's pick a point from the graph. Let's say when \( x = 8 \) (years), \( y = 8 \) (feet). Then \( k=\frac{y}{x}=\frac{8}{8}=1 \). So the equation is \( y = x \). But maybe the problem is to find the height when the tree is a certain age, or the age when the tree is a certain height. Wait, since the problem is about proportional relationship, let's assume we need to find the equation.

But maybe the original problem (since it's a common problem) is to find the constant of proportionality or the equation. Let's proceed with the step - by - step:

Step 1: Recall the formula for proportional relationship

A proportional relationship is given by the equation \( y=kx \), where \( k \) is the constant of proportionality, \( x \) is the independent variable (age of the tree in years), and \( y \) is the dependent variable (height of the tree in feet).

Step 2: Identify a point on the line

Since the graph is a straight line through the origin (because it's a proportional relationship), we can pick any point \( (x,y) \) on the line. Let's choose the point where \( x = 8 \) (years) and \( y = 8 \) (feet) (from the graph, assuming the grid lines are 1 unit apart).

Step 3: Calculate the constant of proportionality \( k \)

Using the formula \( k=\frac{y}{x} \), substitute \( x = 8 \) and \( y = 8 \) into the formula:
\( k=\frac{y}{x}=\frac{8}{8}=1 \)

Step 4: Write the equation of the proportional relationship

Substitute \( k = 1 \) into the equation \( y=kx \…

Step 1: Recall the formula for proportionality

For a proportional relationship \( y = kx \), \( k=\frac{y}{x} \), where \( (x,y) \) is a point on the line.

Step 2: Identify a point on the line

The line passes through \( (8,8) \) (assuming from the graph).

Step 3: Calculate \( k \)

\( k=\frac{y}{x}=\frac{8}{8}=1 \)

Answer:

The constant of proportionality is \( 1 \) (meaning the tree grows 1 foot per year).