he function 𝑆 is given by  𝑆⁡(𝑡)=500,0001+0.4⁢𝑒𝑘⁢𝑡, where 𝑘 is a constant. If 𝑆⁡(4)=300,000, what is the value of 𝑆⁡(12) ?

To solve this problem, we first need to find the value of the constant k. We know that S(4) = 300,000, so we can set up the equation 300,000 = 500,000 / (1 + 0.4e^(4k)).

Step 1: Solve for k 300,000 = 500,000 / (1 + 0.4e^(4k)) Multiply both sides by (1 + 0.4e^(4k)) to get rid of the denominator on the right side: 300,000 * (1 + 0.4e^(4k)) = 500,000 Expand the left side: 300,000 + 120,000e^(4k) = 500,000 Subtract 300,000 from both sides: 120,000e^(4k) = 200,000 Divide both sides by 120,000: e^(4k) = 200,000 / 120,000 = 5/3 Take the natural logarithm of both sides to solve for 4k: 4k = ln(5/3) So, k = ln(5/3) / 4

Step 2: Substitute k into S(t) to find S(12) Now that we have the value of k, we can substitute it into the function S(t) to find S(12): S(12) = 500,000 / (1 + 0.4e^(12k)) Substitute k = ln(5/3) / 4 into the equation: S(12) = 500,000 / (1 + 0.4e^(12 * ln(5/3) / 4)) Solve the equation to find the value of S(12).

Please note that the exact value of S(12) will depend on the value of the natural logarithm of 5/3, which is approximately 0.5108. Therefore, k is approximately 0.5108 / 4 = 0.1277. Substituting this value into the equation for S(12) will give the approximate value of S(12).