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A straight line relationship function also known as a linear function, is one in between the two variables which gives a straight line when the ordered pair are plotted on a graph
The correct values are as follows;
3. a. The cost to go to the fair is $6.00
b. The number of rides Dakota can buy is 18 rides
4. a. The number of gallons of water the fish tank holds is 175 gallons
b. The time it will take the tank to be empty is 17.291[tex]\overline 6[/tex] hours
The reasons the above values are correct are as follows;
3. a, The data in the table are presented as follows;
Number of rides tickets: [tex]{}[/tex] 2, 4, 6, 11
Amount of Money Spent (dollars): 7.5, 9, 10.5, 14.25
From the first three ordered pair values, we have;
The common difference between the amount spent is constant = $1.5, found as follows;
9 - 7.5 = 1.5, 10.5 - 9 = 1.5
Common difference in the number of rides tickets is constant = 2, which is found as follows;
4 - 2 = 2, 6 - 4 = 2
Therefore, the first three sets of values can be represented using a straight line model, y = m·x + c, as follows;
The slope m = (9 - 7.5)/(4 - 2) = 0.75
The equation of the line is, y - 7.5 = 0.75·(x - 2)
y = 0.75·x - 1.5 + 7.5 = 0.75·x + 6
Which gives; y = 0.75·x + 6
When x = 11, we get, y = 0.75 × 11 + 6 = 14.25, therefore, the data is a straight line equation
Therefore, the equation of the data in the table is y = 0.75·x + 6
Where;
y = The amount of money spent
x = The number of rides
Which gives;
The cost of each ride = $0.75
The cost to go to the fair = $6.00
b. The amount Dakota has to spend = $20
To find out the number of rides tickets Dakota can buy, we let the amount of money she spends, y = 20, which gives;
20 = 0.75·x + 6
x = (20 - 6)/0.75 = 18.[tex]\overline 6[/tex]
x = 18.[tex]\overline 6[/tex]
The number of rides Dakota can buy x = 18 rides (the change left 20 - 0.75 × 18 + 6 = 0.5, is not enough for one more ride)
4. a. The given data are presented as follows
Time (hours): [tex]{}[/tex] [tex]\dfrac{1}{4}[/tex], [tex]\dfrac{1}{2}[/tex], [tex]\dfrac{3}{4}[/tex], 1
Amount of Water Remaining (gallons): 169, 163, 157, 151
The common difference between amount of water remaining (gallons) is a constant = -6
The common difference between Time (hours) is a constant = [tex]\dfrac{1}{4}[/tex]
Therefore, the relationship between the gallons remaining and the time is a straight line equation, given as follows;
[tex]y - 169 = \dfrac{ (163 - 169)}{\dfrac{1}{2} - \dfrac{1}{4} } \times \left (x - \dfrac{1}{4} \right) = -24 \cdot x + 6[/tex]
Which gives;
y = -24·x + 6 + 169 = -24·x + 175
y = -24·x + 175
Where;
x = The time
y = The amount of water remaining in the tank in gallons
The amount of water the fish tank originally holds, y is given by when x = 0, as follows;
y = -24 × 0 + 175 = 175
The number of gallons of water the fish tank holds, y = 175 gallons
b. The tank is empty when the number of gallons, y = 0
Plugging in the value of y = 0, in the equation for the water in the fish tank gives;
0 = -24·x + 175
∴ 24·x = 175
x = 175/24 = 17.291[tex]\overline 6[/tex]
The time it will take the tank to be empty, x = 17.291[tex]\mathbf{\overline 6}[/tex] hours
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