Question

# The graph shows population models for three cities , based on data over a five year period. If the populations continue to increase in the same ways , when will the population of city C exceed the populations of the other two cities ?

Hint:

### We observe that-

f(x) is a linear function (highest power is 1),

g(x) is a quadratic function (highest power is 2) and

h(x) is an exponential function (power is a variable).

We will simply substitute different values of x in the given functions and plot the same on a graph and compare the observations.

## The correct answer is: Population of city C will exceed city A in 10 years and city B is approximately 9.5 years.

### Step-by-step solution:-

h(x) = 2^{(x-5)}

Let x = 0- h(x) = 2^{(0-5)} = 2^{(-5)} = 0.03

Let x = 5- h(x) = 2^{(5-5)} = 2^{(0)} = 1

Let x = 12- h(x) = 2^{(12-5)} = 2^{(7)} = 128

∴ We plot the points (0,0.03); (5,1) & (12,128) for h(x)

f(x) = 3.2x

Let x = 0- f(x) = 3.2(0) = 0

Let x = 5- f(x) = 3.2(5) = 16

Let x = 12- f(x) = 3.2(12) = 38.4

∴ We plot the points (0,0); (5,16) & (12,38.4) for f(x)

g(x) = 0.25x^{2}

Let x = 0- g(x) = 0.25(0)^{2} = 0.25 (0) = 0

Let x = 5- g(x) = 0.25(5)^{2} = 0.25 (25) = 6.25

Let x = 12- g(x) = 0.25(12)^{2} = 0.25 (144) = 36

∴ We plot the points (0,0); (5,6.25) & (12,36) for g(x)

From the adjacent graph, we observe that-

Line representing f(x) is a straight line. Hence, f(x) is a linear function.

Line representing g(x) & h(x) are not a straight line. Hence, these are polynomial functions.

However, as increase the x variable, h(x) being an exponential function, increases faster than g(x).

Hence, we see that h(x) will eventually exceed f(x) and g(x).

Also, we observe that h(x) intersects f(x) at the point (10,32) and g(x) at (9.5,22).

Hence, Population of City C will exceed city A in 10 years and city B is approximately 9.5 years.

Final Answer:-

∴ Population of city C will exceed city A in 10 years and city B is approximately 9.5 years.

^{(x-5)}

Let x = 0- h(x) = 2

^{(0-5)}= 2

^{(-5)}= 0.03

Let x = 5- h(x) = 2

^{(5-5)}= 2

^{(0)}= 1

Let x = 12- h(x) = 2

^{(12-5)}= 2

^{(7)}= 128

f(x) = 3.2x

Let x = 0- f(x) = 3.2(0) = 0

Let x = 5- f(x) = 3.2(5) = 16

Let x = 12- f(x) = 3.2(12) = 38.4

∴ We plot the points (0,0); (5,16) & (12,38.4) for f(x)

g(x) = 0.25x

^{2}

Let x = 0- g(x) = 0.25(0)

^{2}= 0.25 (0) = 0

Let x = 5- g(x) = 0.25(5)

^{2}= 0.25 (25) = 6.25

Let x = 12- g(x) = 0.25(12)

^{2}= 0.25 (144) = 36

∴ We plot the points (0,0); (5,6.25) & (12,36) for g(x)

From the adjacent graph, we observe that-

Line representing f(x) is a straight line. Hence, f(x) is a linear function.

Line representing g(x) & h(x) are not a straight line. Hence, these are polynomial functions.

However, as increase the x variable, h(x) being an exponential function, increases faster than g(x).

Hence, we see that h(x) will eventually exceed f(x) and g(x).

Also, we observe that h(x) intersects f(x) at the point (10,32) and g(x) at (9.5,22).

Hence, Population of City C will exceed city A in 10 years and city B is approximately 9.5 years.

Final Answer:-

∴ Population of city C will exceed city A in 10 years and city B is approximately 9.5 years.