Chapter 4 Represent And Solve Equations And Inequalities
Section 4.8: Understand Dependent And Independent Variables
Page 227 Exercise 1a Answer
Three factors that will affect the weight of the box are:
The dimensions of the box. Larger dimensions will weigh more than smaller dimensions.
The contents of the box. Heavier contents will give a larger total weight than lighter contents. The weight of the contents of the box will change based on what present is in the box and how much/what kind of packing material is used to protect the fragile item.
The material used to make the box. Thicker cardboard will weigh more than thinner cardboard.
Three factors that will not affect the weight of the box are:
The orientation of the box. Rotating the box so a different side is on the bottom will not change its weight.
The location of the stickers on the box.
The placement of the contents inside of the box. Placing the item in the middle or closer to one side will not change the overall weight of the box, it will just make the box unbalanced.
Result
Three factors that will affect the weight of the box are the dimensions of the box, the contents of the box, and the material used to make the box. Three factors that will not affect the weight of the box are the orientation of the box, the location of the stickers on the box, and the placement of the contents inside of the box.
Read And Learn More: enVisionmath 2.0 Grade 6 Volume 1 Solutions
Page 227 Exercise 1b Answer
The size of the box affects the weight of the box because a larger box means more cardboard and more tape for the box. The contents of the box affect the weight because a heavier present will increase the total weight of the box. Also, the packing material used inside the box to protect the fragile item will affect the weight. Some packing materials weigh more than others and a larger box would also need more packing materials than a smaller box.
Page 227 Exercise 1 Answer
Possible answer:
Another situation in which changing one factor results in changed to another factor could be wrapping a present. In this situation, the contents of the box would not affect the amount of wrapping paper needed but the size of the box would. Increasing the dimensions of the box would mean the amount of wrapping paper needed would increase.
Result
Possible answer: When wrapping a present, changing the size of the box will affect how much wrapping paper is needed.
Page 228 Exercise 1 Answer
When the baker is making the pancakes, the amount of batter used affects the number of pancakes that can be made. Using more batter means the baker can make more pancakes. Decreasing the amount of batter decreases the number of pancakes that can be made.
The variable p pancakes is then the dependent variable since it changes in response to the number of cups of batter b that are used.
If the baker doubled the number of cups of batter used, I would expect the number of pancakes to double since the number of cups of batter used determines how many pancakes are used.
Result
The variable p pancakes is the dependent variable since it changes in response to the number of cups of batter b that are used. If the baker doubled the number of cups of batter used, I would expect the number of pancakes to double since the number of cups of batter used determines how many pancakes are used.
Page 229 Exercise 2 Answer
From Example 2, we know the independent variables are the length of the rental l, the price per day d, and the price per week w.
Result
The independent variables are the length of the rental l, the price per day d, and the price per week w.
Page 230 Exercise 1 Answer
When a variable is dependent on another variable, that is the dependent variable depends on the independent variable, when the independent variable changes, the dependent variable also changes according to a particular rule.
Result
When the independent variable changes, teh dependent variable also changes.
Page 230 Exercise 2 Answer
The more miles they bike, the more calories they burn. That is, when m, the number of miles cycled, changes, so does c, the number of calories burnt.
Thus, Viola is right. The number of calories, c, depends on the number of miles cycled, m.
Result
Yes.
Page 230 Exercise 3 Answer
In this problem, the dependent variable is the number of calories burnt in an hour, c.
It could depend on how fast they cycle. If they cycle faster, they would burn more calories.
Result
Possible answer: The number of calories burnt could also depend on how fast they cycle. If they cycle faster, they would burn more calories.
Page 230 Exercise 4 Answer
The dependent variable is the amount of money, m, since it depends on the amount of tickets which are sold, t – the independent variable.
Result
m is the dependent variable and t the independent.
Page 230 Exercise 5 Answer
Since the amount of money is greater the more hours one works, the amount of money, m, is the dependent variable and the number of hours worked, h, is the independent variable.
Result
m is the dependent variable and h the independent.
Page 230 Exercise 6 Answer
The more shelves the bookcase has, the more books it can hold. Thus, the number of books, b, the bookcase can hold is the dependent variable and the number of shelves, s, is the independent variable.
Result
b is the dependent variable and s the independent.
Page 230 Exercise 7 Answer
The more hours you read, the more pages you will read. Thus, the number of pages read, p, is the dependent variable and the hours you spent reading, h, is the independent variable.
Result
p is the dependent variable and h is the independent.
Page 230 Exercise 8 Answer
The longer the water runs, the more gallons the hose will produce. Thus, the number of gallons, g, is the dependent variable since it depends on the number of the minutes, m, the water is running, which is the independent variable.
Result
g is the dependent variable and m is the independent.
Page 230 Exercise 9 Answer
The number of peaches is greater the more bushes there are. Thus, the number of peaches, y, is the dependent variable and the number of bushes, x, is the independent variable.
Result
y is the dependent variable and x is the independent.
Page 230 Exercise 10 Answer
The faster you drive the faster you will arrive, that is the number of hours will be less. Thus, the number of hours, h, is the dependent variable and the speed, r, is the independent variable.
Result
h is the dependent variable and r is the independent.
Page 230 Exercise 11 Answer
The more hours the lights are on, the greater the electric bill will be. Also, the greater the price of the electricity, the greater the bill will be.
Thus, one of the independent variable that can change the monthly electric bill is the number of hours one spends with the lights on and the other such variable is the price of electricity.
Result
The number of hours one spends with the lights on and the price of electricity result in a change in a monthly electric bill.
Page 231 Exercise 12 Answer
The more pages the book has, the more it weighs. Thus, the weight, w, is the dependent variable, and the number of pages, p, in a book is the independent variable.
Result
w is the dependent variable and p is the independent.
Page 231 Exercise 13 Answer
The more hamburgers are sold, the greater the amount of money earned. Thus, the amount of sales, s, is the dependent variable and the number of hamburgers, h, is the independent variable.
Result
s is the dependent variable and h is the independent.
Page 231 Exercise 14 Answer
It is implied that you already know the number of bread loaves you want to make and than go buy flour. The more bread loaves you want to make, the more pounds of flour you will have to buy.
Thus, the number of pounds of flour you buy, p, is the dependent variable and the number of bread loaves, b, is the independent variable.
Result
p is the dependent variable and b is the independent.
Page 231 Exercise 15 Answer
The longer the water is in the freezer, the lower its temperature is.
Thus, the temperature, t, is the dependent variable, and the number of minutes the water is in the freezer, m, is the independent variable.
Result
t is the dependent variable and m is the independent.
Page 231 Exercise 16 Answer
Possible answer:
The more you study for a test, the better your test score will be. This means your test score depends on how many hours you study. The independent variable is then the number of hours, h, that you study and the dependent variable is your test score, s.
Result
Possible answer: The more you study for a test, the better your test score will be so the independent variable is the number of hours, h, that you study and the dependent variable is your test score, s.
Page 231 Exercise 17 Answer
The more apples there are in a basket, the more the basket of apples will cost.
The greater the price of one apple, the greater the price of a basket of apples.
Independent variables that could result in a change of the price of a basket of apples are: the number of apples in a basket and the price of one apple.
Result
The number of apples in a basket and the price of one apple.
Page 231 Exercise 18 Answer
The total amount you spend on pants depends on the number of pants you purchase. This means the cost c dollars is the dependent variable and the number of pairs of pants p that you buy is the independent variable. Your friend is then wrong.
Result
Your friend is wrong. The total cost depends on the number of pants purchased so the cost c dollars is the dependent variable and the number of pairs of pants p is the independent variable.
Page 231 Exercise 19 Answer
The more oranges there are in a bag, the more the bag of oranges will cost. Thus, the independent variable is the number of oranges in a bag.
Result
The number of oranges in a bag.
Page 231 Exercise 20 Answer
The growth of a plant depends on the amount of light the plant gets and the amount of nutrients and water the plant gets.
The growth of a plant could also be tracked in terms of time, such as daily or weekly.
Thus, the independent variables in this situation are the amount of light, the amount of nutrients and water, and the amount of time.
Result
The independent variables in this situation are the amount of light, the amount of nutrients and water, and the amount of time.
Page 232 Exercise 21 Answer
We know that distance = (rate)(time). The total distance they drive each day is then affected by their rate (the speed they are driving) and the time (the number of hours they drive).
Possible independent variables that would affect the total distance they drove are then their speed and the number of hours they drive per day.
Result
Their speed and the number of hours they drive per day are two possible independent variables that would affect the total distance they drove each day.
Page 232 Exercise 22 Answer
Some of the factors that would affect the amount of money the Williams family spends on meals during their vacation are the number of people in the family, the total cost for the parents, the total cost for the children, the number of days of the trip, and the cost for each meal.
The cost for the children is affected by factors such as how many children are in the family and the average cost per meal for each child. Since the cost for the children depends on another quantity, then the total cost for the children is a dependent variable.
The cost of each meal is affected by factors such as where the family eats, the sizes of the meals, whether it’s from the regular menu or a childrens’ menu, etc. Since the cost of each meal depends on another quantity, then the cost of each meal is a dependent variable.
Result
Two possible dependent variables are the total cost for the children and the cost of each meal.
Page 232 Exercise 23 Answer
Some of the factors that affect the cost of a salad are its size, the total cost of its ingredients, the amount of a specific ingredient used, and what restaurant it is from.
Two possible independent variables that would affect the cost of a salad are then its size and how much lettuce is used.
Result
Two possible independent variables that would affect the cost of a salad are its size and how much lettuce is used.
Page 232 Exercise 24 Answer
We know that distance = (rate)(time) so the distance Julian drives is affected by both rate and time.
Two possible independent variables that could affect the number of days Julian took to make the trip are then his speed (which is the rate in the distance equation) and the number of hours per day that he drove (which is the time in the distance equation).
Result
Two possible independent variables that could affect the number of days Julian took to make the trip are his speed and the number of hours per day that he drove.
Page 232 Exercise 25 Answer
The more incorrect answers a person gives on a math test, the lowest their test score is. The person’s test score then depends on the number of incorrect answers. Therefore, the dependent variable is the math test score.
Result
The dependent variable is the math test score.
Page 232 Exercise 26 Answer
Possible answers:
If you drive a total distance of d miles in t hours at an average speed of 60 miles per hour, then the total distance d you drive depends on the number of hours t. Therefore, t is the independent variable in this situation.
If you drove a total distance of 500 miles at an average speed s for t hours, then the number of hours your trip takes depends on your speed s. Therefore, t is the dependent variable in this situation.
Result
Possible answers:
If you drive a total distance of d miles in t hours at an average speed of 60 miles per hour, then t is the independent variable.
If you drove a total distance of 500 miles at an average speed s for t hours, then t is the dependent variable.
Page 232 Exercise 27a Answer
We are given the following table:
As we can see from the table, the newer machines use fewer gallons of water. One possible independent variable that could affect the amount of water Jonas uses to do the laundry could then be the age of the washing machine.
Also from the table, we can see the more energy-efficient model uses fewer gallons of water so a second possible independent variable could be the amount of energy used to run the machine.
Larger washing machines will require more gallons of water so a third possible independent variable could be the size of the washing machine.
Result
Possible answers: age of the washing machine, the amount of energy used, the size of the washing machine
Page 232 Exercise 27b Answer
The total cost, c, depends on the number of gallons of water, g, that he uses. This is because an increase in the number of gallons will cause the cost to increase.
The dependent variable is then the total cost, c, and the independent variable is the number of gallons of water, g.
Result
The dependent variable is the total cost, c, and the independent variable is the number of gallons of water, g.