Question

amount a and the rate of growth r (as a percent) of the exponential functiound your answer to the nearest tenth.

1. $y = 10(1 + 0.4)^t$
2. $y = 12(1.05)^t$
3. $h(t) = 175(1.028)^t$
4. $p(t) = 1.8^t$

Explanation:

Problem 2: \( y = 10(1 + 0.4)^t \)

Step 1: Recall the exponential growth formula

The general form of an exponential growth function is \( y = a(1 + r)^t \), where \( a \) is the initial amount and \( r \) is the growth rate (as a decimal).

Step 2: Identify \( a \) and \( r \)

For the function \( y = 10(1 + 0.4)^t \), comparing with \( y = a(1 + r)^t \), we have:

• Initial amount \( a = 10 \)
• Growth rate (decimal) \( r = 0.4 \)

Step 3: Convert \( r \) to a percent

To convert the decimal growth rate to a percent, multiply by 100: \( r = 0.4 \times 100 = 40.0\% \)

Step 1: Recall the exponential growth formula

The general form of an exponential growth function is \( y = a(1 + r)^t \), where \( a \) is the initial amount and \( r \) is the growth rate (as a decimal). We can rewrite \( 1.05 \) as \( 1 + 0.05 \).

Step 2: Identify \( a \) and \( r \)

For the function \( y = 12(1.05)^t = 12(1 + 0.05)^t \), comparing with \( y = a(1 + r)^t \), we have:

• Initial amount \( a = 12 \)
• Growth rate (decimal) \( r = 0.05 \)

Step 3: Convert \( r \) to a percent

To convert the decimal growth rate to a percent, multiply by 100: \( r = 0.05 \times 100 = 5.0\% \)

Step 1: Recall the exponential growth formula

The general form of an exponential growth function is \( y = a(1 + r)^t \), where \( a \) is the initial amount and \( r \) is the growth rate (as a decimal). We can rewrite \( 1.028 \) as \( 1 + 0.028 \).

Step 2: Identify \( a \) and \( r \)

For the function \( h(t) = 175(1.028)^t = 175(1 + 0.028)^t \), comparing with \( y = a(1 + r)^t \), we have:

• Initial amount \( a = 175 \)
• Growth rate (decimal) \( r = 0.028 \)

Step 3: Convert \( r \) to a percent

To convert the decimal growth rate to a percent, multiply by 100: \( r = 0.028 \times 100 = 2.8\% \)

Answer:

Initial amount \( a = 10 \), growth rate \( r = 40.0\% \)

Problem 4: \( y = 12(1.05)^t \)