Question
Function f represents the population , in millions , of Franklin x years from now. Function g represents the population , in millions , of Georgetown x years from now, If the pattern shown in the table continue , will franklin always have a greater population than Georgetown ? Explain.
Hint:
1. When the difference between 2 consecutive output values (y values) for a given constant change in the input values (x values) is constant. i.e. y(n)- y(n-1) is constant for any value of n, the function is known as a linear function.
2. When the difference between 2 consecutive differences for output values (y values) for a given constant change in the input values (x values) is constant. i.e. dy(n)- dy(n-1) is constant for any value of n, the function is known as a quadratic function.
3. When the ratio between 2 consecutive output values (y values) for a given constant change in the input values (x values) is constant i.e. y(n)/y(n-1) is constant for any value of n, the function is known as an exponential function.
The correct answer is: Since the population of Franklin is a Linear function and that of Georgetown is an exponential function, Franklin will not have a greater population in the long run.
Step-by-step solution:-
From the given table, we observe the following readings-
x1 = 0, y1(f) = 3.4, y1(g) = 2.4;
x2 = 1, y2(f) = 5.6, y1(g) = 3.6;
x3 = 2, y3(f) = 7.8, y1(g) = 5.4;
x4 = 3, y4(f) = 10, y1(g) = 8.1
Difference between 2 consecutive x values-
dx1 = x2 - x1 = 1 - 0 = 1
dx2 = x3 - x2 = 2 - 1 = 1
dx3 = x4 - x3 = 3 - 2 = 1
a). For Franklin-
Difference between 2 consecutive y values-
dy1 = y2 - y1 = 5.6 - 3.4 = 2.2
dy2 = y3 - y2 = 7.8 - 5.6 = 2.2
dy3 = y4 - y3 = 10 - 7.8 = 2.2
We observe that the difference for every consecutive x values is constant i.e. 1 and for y values the difference is constant i.e. 2.2.
Hence, the given function is a linear function.
b). For Georgetown-
Difference between 2 consecutive y values-
dy1 = y2 - y1 = 3.6 - 2.4 = 1.2
dy2 = y3 - y2 = 5.4 - 3.6 = 1.8
dy3 = y4 - y3 = 8.1 - 5.4 = 2.7
We observe that the difference for every consecutive x values is constant i.e. 1 but for y values the difference is not constant.
Hence, the given function is not a linear function.
Now, ratio between 2 consecutive y values-
= 3.6/2.4 = 1.5
= 5.4/3.6 = 1.5
= 8.1/5.4 = 1.5
We observe that difference between 2 consecutive y values is constant i.e. 1.5.
Hence, the given function is an exponential function.
We know from the above calculations that the population growth of Franklin is a Linear function and that of Georgetown is an exponential function. We also know that for a linear function, the rate of change in output for a given change in input is linear i.e. constant. However, for an exponential function, the rate of change in output for a given change in input keeps increasing at an exponential level. Hence, the population of Georgetown will far exceed that of Franklin in the long run.
Final Answer:-
∴ Since the population of Franklin is a Linear function and that of Georgetown is an exponential function, Franklin will not have a greater population in the long run.
x2 = 1, y2(f) = 5.6, y1(g) = 3.6;
x3 = 2, y3(f) = 7.8, y1(g) = 5.4;
x4 = 3, y4(f) = 10, y1(g) = 8.1
Difference between 2 consecutive x values-
dx1 = x2 - x1 = 1 - 0 = 1
dx2 = x3 - x2 = 2 - 1 = 1
dx3 = x4 - x3 = 3 - 2 = 1
a). For Franklin-
Difference between 2 consecutive y values-
dy1 = y2 - y1 = 5.6 - 3.4 = 2.2
dy2 = y3 - y2 = 7.8 - 5.6 = 2.2
dy3 = y4 - y3 = 10 - 7.8 = 2.2
We observe that the difference for every consecutive x values is constant i.e. 1 and for y values the difference is constant i.e. 2.2.
Hence, the given function is a linear function.
b). For Georgetown-
Difference between 2 consecutive y values-
dy1 = y2 - y1 = 3.6 - 2.4 = 1.2
dy2 = y3 - y2 = 5.4 - 3.6 = 1.8
dy3 = y4 - y3 = 8.1 - 5.4 = 2.7
We observe that the difference for every consecutive x values is constant i.e. 1 but for y values the difference is not constant.
Hence, the given function is not a linear function.
Now, ratio between 2 consecutive y values-
= 3.6/2.4 = 1.5 = 5.4/3.6 = 1.5 = 8.1/5.4 = 1.5We observe that difference between 2 consecutive y values is constant i.e. 1.5.
Hence, the given function is an exponential function.
We know from the above calculations that the population growth of Franklin is a Linear function and that of Georgetown is an exponential function. We also know that for a linear function, the rate of change in output for a given change in input is linear i.e. constant. However, for an exponential function, the rate of change in output for a given change in input keeps increasing at an exponential level. Hence, the population of Georgetown will far exceed that of Franklin in the long run.
Final Answer:-
∴ Since the population of Franklin is a Linear function and that of Georgetown is an exponential function, Franklin will not have a greater population in the long run.
Round to the nearest tenth if necessary.
tells the amount of drink left in your system after t hours. How much of the soft drink will be left in your system after 16 hours?
