This answer key provides solutions and explanations for a hypothetical Lesson 11 on equations for proportional relationships. Since I don't have access to your specific homework assignment, I'll create example problems covering the key concepts typically found in such a lesson. Remember to always refer to your textbook and class notes for the most accurate answers to your specific problems.
Understanding Proportional Relationships
A proportional relationship exists between two variables when their ratio remains constant. This constant ratio is often represented by the constant of proportionality, often denoted as k. We can express proportional relationships using the equation:
y = kx
where:
- y is the dependent variable
- x is the independent variable
- k is the constant of proportionality
Example Problems and Solutions
Here are some example problems and their solutions to illustrate the concepts covered in a typical Lesson 11 on equations for proportional relationships:
Problem 1:
The number of cookies baked (y) is proportional to the number of batches (x) made. If 2 batches yield 24 cookies, find the equation representing this relationship. Then, determine how many cookies are baked from 5 batches.
Solution:
-
Find the constant of proportionality (k):
We know that y = kx. With 2 batches (x=2) yielding 24 cookies (y=24), we can solve for k:
24 = k * 2
k = 24 / 2 = 12
-
Write the equation:
The equation representing the relationship is: y = 12x
-
Determine cookies from 5 batches:
Substitute x = 5 into the equation:
y = 12 * 5 = 60
Therefore, 5 batches yield 60 cookies.
Problem 2:
The distance (y) traveled by a car is proportional to the time (x) spent driving at a constant speed. If the car travels 150 miles in 3 hours, what is the equation representing this relationship? How far will the car travel in 7 hours?
Solution:
-
Find k:
150 = k * 3
k = 150 / 3 = 50 miles per hour (this is the speed)
-
Write the equation:
y = 50x
-
Distance in 7 hours:
y = 50 * 7 = 350 miles
Problem 3:
The cost (y) of apples is proportional to the weight (x) of apples purchased. If 2 pounds of apples cost $4, what is the equation relating cost and weight? What would be the cost of 5 pounds of apples?
Solution:
-
Find k:
4 = k * 2
k = 4 / 2 = $2 per pound
-
Write the equation:
y = 2x
-
Cost of 5 pounds:
y = 2 * 5 = $10
Problem 4 (Graphing):
If you are given a graph showing a proportional relationship, how can you find the equation?
Solution:
Find any point (x, y) on the graph. The constant of proportionality, k, is the slope of the line, which can be calculated as k = y/x. The equation will always be in the form y = kx.
Remember to always check your work and ensure your answers make sense within the context of the problem. If you have specific problems from your assignment, please provide them, and I will do my best to help you solve them.
