Question

laundry
b. describe how the unit rate is represented in the graph.

1. open response daniella makes apple pies each fall. the cost at the local grocery store for x pounds of apples is shown in the table. what is the least amount of money daniella will spend for 15 pounds of apples? assume the relationship is proportional. (lesson 1)

number of pounds, x | total cost ($), y
2 | $4.50
3 | $6.75

1. open response the points in the table lie on a line. compute the slope of the line. (lesson 2)

x | y
6 | -3
-2 | 1
-4 | 2

1. multiselect which statement is true about the graph? select all that apply. (lesson 3)

the ratio of the rise to the run of each triangle is the same.
the smaller triangle and the larger triangle shown are similar.
the slope of the line is 2.
the slope of the line is -2.
the corresponding sides of the two triangles are not proportional.
module 4 • linear relationships and slope 249

Explanation:

Problem 2 (Open Response: Cost for 15 pounds of apples)

Step1: Find the unit rate (cost per pound)

Since the relationship is proportional, we can use the formula for a proportional relationship \( y = kx \), where \( k \) is the constant of proportionality (unit rate). Using the table values, when \( x = 2 \), \( y = 4.50 \). So, \( k=\frac{y}{x}=\frac{4.50}{2}=2.25 \) dollars per pound.

Step2: Calculate cost for 15 pounds

Now, use the proportional relationship \( y = kx \) with \( x = 15 \) and \( k = 2.25 \). So, \( y=2.25\times15 = 33.75 \).

Step1: Recall the slope formula

The slope \( m \) between two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take two points from the table, say \( (6, - 3) \) and \( (-2,1) \).

Step2: Apply the slope formula

Substitute \( x_1 = 6,y_1=-3,x_2=-2,y_2 = 1 \) into the formula: \( m=\frac{1-(-3)}{-2 - 6}=\frac{4}{-8}=-\frac{1}{2} \). (We can verify with another pair, e.g., \( (-2,1) \) and \( (-4,2) \): \( m=\frac{2 - 1}{-4-(-2)}=\frac{1}{-2}=-\frac{1}{2} \), same result.)

• "The ratio of the rise to the run of each triangle is the same": Since the triangles are formed by the line, their rise - run ratios (slope) are equal (as slope is constant for a line). So this is true.
• "The smaller triangle and the larger triangle shown are similar": Similar triangles have equal corresponding angles and proportional sides. Since their rise - run ratios (slope) are equal, the angles are equal, so they are similar. This is true.
• "The slope of the line is 2": The line is decreasing, so slope should be negative. So this is false.
• "The slope of the line is - 2": Let's check. If we take two points, the rise over run is negative, but from the graph, if we consider the change in \( y \) and \( x \), the slope is \( - 2 \)? Wait, no, let's re - evaluate. Wait, if we look at the graph, when \( x \) increases by 1, \( y \) decreases by 2? Wait, no, maybe my initial thought was wrong. Wait, actually, from the graph, if we take two points on the line, the slope calculation: Let's say the line goes from a point to another. But the key points:
• The ratio of rise to run (slope) is the same for both triangles (since they are on the same line), so first statement is true.
• The triangles are similar (same angles, as slope is same, so corresponding angles equal), so second statement is true.
• Slope is negative (line is decreasing), so "slope is 2" is false. "Slope is - 2": Let's check with the graph. If we take two points, say when \( x = 0 \), \( y \) is some value, and when \( x \) increases by 1, \( y \) decreases by 2? Maybe. But the first two statements are true, and "the corresponding sides of the two triangles are not proportional" is false (since they are similar, sides are proportional).

Answer:

\( \$33.75 \)

Problem 4 (Open Response: Slope of the line)