The radioactive substance uranium-240 has a half-life of 14 hours. The amount $A(t)$ of a sample of uranium-240 remaining (in grams) after $t$ hours is given by the following exponential function. \[ A(t)=4300\left(\frac{1}{2}\right)^{\frac{t}{14}} \] Find the amount of the sample remaining after 9 hours and after 40 hours. Round your answers to the nearest gram as necessary. Amount after 9 hours: grams Amount after 40 hours: grams Explanation Check - 2023 McGraw Hill LLC. All Rights Reserved. Terms of Use I Privacy Center I Accessibility.

Step 1 ：The radioactive substance uranium-240 has a half-life of 14 hours. The amount $A(t)$ of a sample of uranium-240 remaining (in grams) after $t$ hours is given by the following exponential function.

Step 2 ：\[A(t)=4300\left(\frac{1}{2}\right)^{\frac{t}{14}}\]

Step 3 ：We are asked to find the amount of the sample remaining after 9 hours and after 40 hours. We can use the given exponential function to calculate this. We just need to substitute the values of t (time in hours) into the function and calculate the result.

Step 4 ：For 9 hours, we substitute $t=9$ into the function to get:

Step 5 ：\[A_9 = 4300\left(\frac{1}{2}\right)^{\frac{9}{14}}\]

Step 6 ：Calculating this gives $A_9 = 2753.906382993186$ grams. Rounding to the nearest gram gives $A_9 = 2754$ grams.

Step 7 ：For 40 hours, we substitute $t=40$ into the function to get:

Step 8 ：\[A_{40} = 4300\left(\frac{1}{2}\right)^{\frac{40}{14}}\]

Step 9 ：Calculating this gives $A_{40} = 593.4481135996741$ grams. Rounding to the nearest gram gives $A_{40} = 593$ grams.

Step 10 ：Final Answer: The amount of the sample remaining after 9 hours is \(\boxed{2754}\) grams and after 40 hours is \(\boxed{593}\) grams.