Question

The amount of carbon dioxide (CO₂) in the atmosphere increases rapidly as we continue to rely on fossil fuels. The relationship between the elapsed time, t, in decades, since CO₂ levels were first measured, and the total amount of CO₂ in the atmosphere, a_decade(t), in parts per million, is modeled by the following function: a_decade(t) = 315 * (1.06)⁽ᵗ⁾. Complete the following sentence about the yearly rate of change in the amount of CO₂ in the atmosphere. Round your answer to four decimal places. Every year, the amount of CO₂ in the atmosphere increases by a factor of

Answer 1

The yearly rate of change in the amount of CO₂ in the atmosphere increases by a factor of 1.06.

Step-by-step explanation:

The yearly rate of change in the amount of CO₂ in the atmosphere can be determined by finding the ratio of the amount of CO₂ in a given year to the amount of CO₂ in the previous year.

In this case, the function a_decade(t) = 315 * (1.06)⁽ᵗ⁾ represents the total amount of CO₂ in the atmosphere in parts per million (ppm) as a function of the elapsed time in decades, t, since CO₂ levels were first measured. To find the yearly rate of change, we need to find the ratio of a_decade(t+1) to a_decade(t), as t represents time in decades.

Let's calculate the ratio:

a_decade(t+1) / a_decade(t) = (315 * (1.06)⁽ᵗ⁺¹⁾) / (315 * (1.06)⁽ᵗ⁾)

Simplifying the expression:

a_decade(t+1) / a_decade(t) = 1.06

Therefore, every year, the amount of CO₂ in the atmosphere increases by a factor of 1.06.