Done myopenmath.com Submit Question Question 4 1.5/3 pts \( 3 \stackrel{38}{\rightleftarrows} \) Details Certain radioactive material decays in such a way that the mass remaining after \( t \) years is given by the function \[ m(t)=175(0.965)^{t} \] where \( m(t) \) is measured in grams. (a) Find the mass at time \( t=0 \). Your answer is \( \square \) (b) How much of the mass remains after 40 years? Your answer is \( \square \) Give your answer to two decimal places. Submit Question Question 5 5/5 pts \( 2 \underset{ }{\rightleftarrows} 99 \) Details

Sure, let's work through both parts of **Question 4** step by step. --- ### **Question 4** The mass remaining after \( t \) years is given by the function: \[ m(t) = 175 \times (0.965)^t \] where \( m(t) \) is measured in grams. --- #### **(a) Find the mass at time \( t = 0 \).** **Solution:** To find the mass at \( t = 0 \), simply substitute \( t = 0 \) into the function: \[ m(0) = 175 \times (0.965)^0 \] Recall that any number raised to the power of 0 is 1: \[ m(0) = 175 \times 1 = 175 \text{ grams} \] **Answer:** \( 175 \) grams --- #### **(b) How much of the mass remains after 40 years?** **Solution:** To find the mass after 40 years, substitute \( t = 40 \) into the function: \[ m(40) = 175 \times (0.965)^{40} \] First, calculate \( (0.965)^{40} \): \[ (0.965)^{40} \approx 0.2418 \] Now, multiply by 175 grams: \[ m(40) = 175 \times 0.2418 \approx 42.315 \text{ grams} \] Rounded to two decimal places: \[ m(40) \approx 42.32 \text{ grams} \] **Answer:** \( 42.32 \) grams --- Feel free to ask if you need further clarification on any of the steps!