Question

Each decade, the Cloud Peak city council conducts a census to determine the city's population. According to this year's census, the city has a population of 143,501 residents. After analyzing the census data, the council members believe that Cloud Peak's population will decrease by about 7% each decade. Write an exponential equation in the form y=a(b)ˣ that can model the population, y, x decades after this census was taken. Use whole numbers, decimals, or simplified fractions for the values of a and b.

y=____
After how many decades will the population of Cloud Peak fall below 100,000 residents?

Answer 1

To model the population of Cloud Peak after each decade, use the exponential equation y = a(b)^x, where y represents the population, x represents the number of decades, and b represents the decrease rate. The population of Cloud Peak is expected to decrease by about 7% each decade. To find out after how many decades the population will fall below 100,000 residents, solve the equation y = 100,000 and find the value of x.

Step-by-step explanation:

To model the population of Cloud Peak after x decades, we can use the exponential equation y = a(b)^x, where y represents the population and x represents the number of decades. In this case, the population is expected to decrease by about 7% each decade, so the base, b, of the exponential equation would be 0.93 (1 - 0.07). We can assume the initial population, y, from the current census is 143,501.

The exponential equation becomes y = 143,501(0.93)^x.

To find out after how many decades the population of Cloud Peak will fall below 100,000 residents, we need to solve the equation y = 100,000 and find the value of x.

100,000 = 143,501(0.93)^x

Divide both sides by 143,501: 0.6966 ≈ (0.93)^x

Take the logarithm of both sides: log 0.6966 ≈ x log 0.93

Using a logarithm calculator, we find that x is approximately 8.796 decades. Therefore, it will take approximately 8.8 decades for the population of Cloud Peak to fall below 100,000 residents.