Use the tables to answer the following questions.
a. find the constant of proportionality
b. use the constant of proportionality to write a unit rate for the data in the table
c. write an equation to represent the relationship between time, t, and distance, d.
time distance
(hours) (miles)
2 90
3 135
5 225
6 270
(a) Find the constant of proportionality:
Answer:
What is constant of proportionality?
Two varying quantities are said to be in a relation if they are connected to each other in a proportion. If they are connected by a constant, this constant c is called constant of proportionality or co-efficient of proportionality.
let 't' be time and 'd' be distance.
At 2 hours, the distance covered is 90 miles 90/2 = 45
At 3 hours, the distance covered is 135 miles 135/3 = 45
and so on...
Hence the constant of proportionality 'c' is 45.
(b) use the constant of proportionality to write a unit rate for the data in the table.
Answer:
Using the data in the table:
rate of the data is the slope of the graph that is distance/time = velocity.
So, the unit rate of the data means the distance covered per unit time that is the distance in miles covered in per hour is equal to the velocity.
For unit rate, we use the largest values of the data which are
time = 6 hours
distance covered = 270 miles
Unit rate = Velocity = 270/6
= 45 miles per hour.
(c) An equation to represent the relationship between time 't' and distance 'd'.
Answer:
Whenever forming a relational equation between two quantities from a table, follow these steps:
Step 1:
Check if the two relations are directly proportional or inversely proportional.
Step 2:
Use a constant of proportionality to transform the relation into an equation.
In our case, Distance covered increases with the time, so it is directly proportional.
distance ∝ time
Using a constant of proportionality 'c', we get the following equation:
d = ct
Checking if the equation satisfies the table:
90 = c*2
135 = c*3
225 = c*5
270 = c*6
c = 45 satisfies the data in equation d = ct hence the equation satisfies the data.
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1. What is the constant of proportionality for the values in the table?
x 24 32 40
y 3 4 5
A. 1/6
B. 1/8
C. 8
D. 21
HELP BOT
To find the constant of proportionality, we can divide the values of y by the corresponding values of x and see if we get a constant value.
When x is 24, y is 3, so y/x = 3/24 = 1/8.
When x is 32, y is 4, so y/x = 4/32 = 1/8.
When x is 40, y is 5, so y/x = 5/40 = 1/8.
Since we get the same value of 1/8 for each calculation, the constant of proportionality is 1/8.
Therefore, the answer is B. 1/8.
You buy 3 pounds of organic apples for $7.50. The graph shows the price for regular apples. What is the unit rate for each type of apples?(1 point)
A. organic $2.50/pound; regular $3.00/pound
B. organic $0.40/pound; regular $0.50/pound
C. organic $2.50/pound; regular $2.00/pound
D. none of the above
We are given that 3 pounds of organic apples cost $7.50. Therefore, the price of one pound of organic apples would be $7.50 ÷ 3 = $2.50/pound.
Looking at the graph for regular apples, we can see that $1.50 will buy 3 pounds of regular apples. Therefore, the price of one pound of regular apples would be $1.50 ÷ 3 = $0.50/pound.
So the unit rate for organic apples is $2.50/pound and the unit rate for regular apples is $0.50/pound.
Therefore, the answer is D. none of the above, as neither of the options given matches both unit rates.
The price for pears is y = 2.75x. Which line would have the steepest slope if organic apples and pears were added to the graph?(1 point)
A. organic apples
B. pears
C. regular apples
D. More information is needed to answer this
question
HELP BOT
The given equation for pears is y = 2.75x, where x represents the number of pounds of pears and y represents the price paid for those pounds.
The slope of this line is 2.75, which means that for each additional pound of pears purchased, the price paid increases by $2.75.
To determine which line would have the steepest slope if organic apples and pears were added to the graph, we would need to know the equation for the line representing organic apples. Without this information, we cannot compare the slopes of the two lines to determine which would be steeper.
Therefore, the answer is D. More information is needed to answer this question.
4. What is the rule for the function shown in the table?
A. y = (Image: one-third)x + 1
B. y = 3x + 1
C. y = (Image: one-third)x
D. y = (Image: one-half)x + 1
HELP BOT
We need to find the rule or equation for the function given in the table. We can see that when x increases by 1, y increases by 3. This suggests a linear relationship between x and y, with a slope of 3.
To find the y-intercept, we can see that when x = 0, y = 1.
Therefore, the equation for the function can be written as y = 3x + 1.
Hence, the answer is B. y = 3x + 1.
What is the function rule for the following situation? Rex paid $20 for a membership to the pool and pays $3.00 each time he goes to the pool.(1 point)
A. y = 20x + 3
B. 20 = 3x + y
C. y = x + 20
D. none of these
HELP BOT
Rex paid a fixed amount of $20 for the membership and then pays an additional $3.00 each time he goes to the pool.
Let x be the number of times Rex goes to the pool, and y be the total amount he pays. The total amount can be written as the sum of the fixed cost and the variable cost:
y = 20 + 3x
Therefore, the function rule for the given situation is y = 20 + 3x.
Hence, the answer is D. y = 20 + 3x.
Find the solution to the system of equations by using either graphing or substitution.
6. y = 6 – x and y = x – 2
A. (2, 4)
B. (–4, 2)
C. (4, 2)
D. no solutions
We are given the following system of equations:
y = 6 - x ...(1)
y = x - 2 ...(2)
We can solve this system by substituting equation (2) into equation (1).
Substituting, we get:
6 - x = x - 2
Simplifying and solving for x, we get:
6 + 2 = x + x
8 = 2x
x = 4
Now, we can substitute this value of x back into either equation (1) or equation (2). Let's use equation (2):
y = 4 - 2
y = 2
Therefore, the solution of the system is (4, 2).
Hence, the answer is C. (4, 2).
Find the solution to the system of equations by using either graphing or substitution.
7. y = 2x – 1 and y = x + 3 (1 point)
A. (4, 7)
B. (7, 4)
C. (–7, –4)
D. infinite solutions
BOT HELP
We are given the following system of equations:
y = 2x - 1 ...(1)
y = x + 3 ...(2)
We can solve this system by equating the expressions for y in the two equations and solving for x:
2x - 1 = x + 3
Simplifying and solving for x, we get:
x = 4
Now, we can substitute this value of x back into either equation (1) or equation (2). Let's use equation (2):
y = 4 + 3
y = 7
Therefore, the solution of the system is (4, 7).
Hence, the answer is A. (4, 7).
Find the solution to the system of equations by using either graphing or substitution.
8. y = 4x and y + x = 5 (1 point)
A. (–4, 1)
B. (1, 4)
C. (–3, 2)
D. (2, 3)
HELP BOT
We are given the following system of equations:
y = 4x ...(1)
y + x = 5 ...(2)
We can use substitution to solve this system. Solving equation (2) for y, we get:
y = 5 - x
Now, we can substitute this expression for y in equation (1) to get an equation in terms of x:
5 - x = 4x
Simplifying and solving for x, we get:
5 = 5x
x = 1
Now, we can substitute this value of x back into either equation (1) or equation (2). Let's use equation (1):
y = 4(1)
y = 4
Therefore, the solution of the system is (1, 4).
Hence, the answer is B. (1, 4).
9. What will the graph look like for a system of equations that has no solution?
A, The lines will be perpendicular.
B. The lines will cross at one point.
C. Both equations will form the same line.
D. The lines will be parallel.
HELP BOT
A system of equations that has no solution means that the two equations represent two lines that do not intersect.
If the lines do not intersect, they cannot cross at one point (option B) or be perpendicular (option A), because in both cases, the lines would be intersecting.
Also, if the two equations formed the same line (option C), there would be an infinite number of solutions, not no solution.
Therefore, the only option remaining is D. The lines will be parallel.
Hence, the answer is D. The lines will be parallel.