Question

Match each exponential function with its type and constant percent rate of change.(1 point)Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.y=270(0.5)0.7ty=400(1.04)t2y=42(1.67)t5y=640(0.83)0.65tgrowth rate of about 2%growth rate of about 11%decay rate of about 11%decay rate of about 38%

Answer 1

To match each exponential function with its type and constant percent rate of change, let's analyze each function:

1. \( y = 270(0.5)^{0.7t} \)

   • This function represents exponential decay because the base (0.5) is less than 1.
   • The percent rate of change can be calculated:
     • Rate of change = \( (1 - 0.5) \times 100 = 50% \)
     • The decay rate here can roughly correspond to 38% (because \( 0.5 \) decays at a rate greater than 38% over shorter time intervals).
   • Matched with: Decay rate of about 38%.

2. \( y = 400(1.04)^{2t} \)

   • This function represents exponential growth as the base (1.04) is greater than 1.
   • The percent rate of change is:
     • Rate of change = \( (1.04 - 1) \times 100 = 4% \) per unit time.
     • Since the base is raised to \( 2t \), it effectively doubles the rate: \( 4% \times 2 = 8% \) per \( t \), but still matches the immediate growth response as growth rates are applied.
   • Matched with: Growth rate of about 11%.

3. \( y = 42(1.67)^{5t} \)

   • This also represents exponential growth (1.67 > 1).
   • The percent rate of change is:
     • Rate of change = \( (1.67 - 1) \times 100 = 67% \) per unit time.
     • The function has a significant growth factor, suggesting a considerable increase.
   • Matched with: Growth rate of about 11%. (Note that here it exceeded, so correct data is matched before).

4. \( y = 640(0.83)^{0.65t} \)

   • This function represents exponential decay because the base (0.83) is less than 1.
   • The percent rate of change:
     • Rate of change = \( (1 - 0.83) \times 100 = 17% \) decay for the unit, but lower over time factor across 0.65t multiplies decay at a reduced pace.
   • Matched with: Decay rate of about 11%.

So here are the final matches based on the analysis:

1. \( y = 270(0.5)^{0.7t} \) - Decay rate of about 38%
2. \( y = 400(1.04)^{2t} \) - Growth rate of about 11%
3. \( y = 42(1.67)^{5t} \) - Growth rate of about 11%
4. \( y = 640(0.83)^{0.65t} \) - Decay rate of about 11%

Please double check the rates based on correct growth or decay for precise matching if required on notation.