Let's break down the problem and solve it step by step. ### **Given Information** - Initial area, \( y_0 = 64.7 \) mm² - Doubling time = 21 minutes Let: - \( t \) = time in minutes - \( y \) = area at time \( t \) The area grows **exponentially** with a certain doubling time. --- ## **Step (a): Write the exponential formula** The general formula for exponential growth is: \[ y = y_0 \cdot b^{t} \] But since we know the doubling time, we use: \[ y = y_0 \cdot 2^{t/T} \] where: - \( T \) = doubling time (21 minutes) So, \[ y = 64.7 \cdot 2^{t/21} \] **This is the required formula.** --- ## **Step (b): Find the area after 18 minutes** Plug \( t = 18 \) into the formula: \[ y = 64.7 \cdot 2^{18/21} \] Now, calculate \( 2^{18/21} \): First, \( \frac{18}{21} = \frac{6}{7} \). So, \[ y = 64.7 \cdot 2^{6/7} \] Calculate \( 2^{6/7} \) (using a calculator): \[ 2^{6/7} \approx 1.811 \] Now multiply: \[ y \approx 64.7 \times 1.811 \approx 117.1 \] --- ### **Final Answers** **(a) Formula:** \[ y = 64.7 \cdot 2^{t/21} \] where \( t \) is in minutes. **(b) Area after 18 minutes:** \[ \boxed{117.1\ \text{mm}^2} \] (rounded to the nearest tenth)

Certainly! Let's go through each part with detailed calculations step-by-step, using the given data. --- ### **Given Data:** - Initial area, \( y_0 = 64.7 \text{ mm}^2 \) - Doubling time, \( T = 21 \text{ minutes} \) --- ## **Part (a): Write the exponential growth formula** ### Step 1: Understand the nature of growth Since the area doubles every \( T = 21 \) minutes, the exponential growth model can be written as: \[ y(t) = y_0 \times 2^{t/T} \] Where: - \( y(t) \) = area at time \( t \) - \( y_0 \) = initial area - \( t \) = time in minutes - \( T \) = doubling time ### Step 2: Write the formula explicitly Plugging the known values: \[ \boxed{ \textbf{(a)}\quad y(t) = 64.7 \times 2^{t/21} } \] This formula gives the area \( y \) at any time \( t \) in minutes. --- ## **Part (b): Find the area after 18 minutes** ### Step 1: Substitute \( t = 18 \) minutes into the formula \[ y(18) = 64.7 \times 2^{18/21} \] ### Step 2: Simplify the exponent \[ \frac{18}{21} = \frac{6}{7} \quad \text{(since 18 ÷ 3 = 6, 21 ÷ 3 = 7)} \] So, \[ y(18) = 64.7 \times 2^{6/7} \] ### Step 3: Calculate \( 2^{6/7} \) **Method:** Use logarithms or a calculator. - First, recall that: \[ 2^{6/7} = e^{(6/7) \times \ln 2} \] - Since \( \ln 2 \approx 0.6931 \): \[ 2^{6/7} = e^{(6/7) \times 0.6931} \] - Calculate the exponent: \[ (6/7) \times 0.6931 \approx 0.5929 \] - Now, compute \( e^{0.5929} \): \[ e^{0.5929} \approx 1.809 \] **Alternatively**, using a calculator directly: \[ 2^{6/7} \approx 1.809 \] ### Step 4: Calculate the area \[ y(18) = 64.7 \times 1.809 \approx 117.1 \] ### **Final answer:** \[ \boxed{ \text{Area after 18 minutes} \approx \mathbf{117.1\ \text{mm}^2} } \] --- ## **Summary:** | Part | Formula / Calculation | Final Result | |---------|-------------------------|--------------| | (a) | \( y = 64.7 \times 2^{t/21} \) | - | | (b) | \( y(18) = 64.7 \times 2^{6/7} \approx 64.7 \times 1.809 \) | **117.1 mm²** | --- Let me know if you'd like further explanation or help with other parts!

**Part (a):** The exponential growth formula is: \[ \boxed{ y = 64.7 \times 2^{t/21} } \] where \( t \) is in minutes. **Part (b):** Area after 18 minutes: \[ y(18) = 64.7 \times 2^{6/7} \approx 64.7 \times 1.809 \approx \boxed{117.1\ \text{mm}^2} \]