Questions: Now, remembering that ln(x^n) = n ln(x) and that e^(ln(z)) = z, we have y(t) = 30 e^(t / 30)(-ln(2)) = mg.

Transcript text: Now, remembering that $\ln \left(x^{n}\right)=n \ln (x)$ and that $e^{\ln (z)}=z$, we have \[ y(t)=30 e^{(t / 30)(-\ln (2))}= \] $\square$ mg.

Solution

Solution Steps

To solve for $ y(t) $, we need to simplify the expression using the properties of logarithms and exponents. The expression given is $ y(t) = 30 e^{(t / 30)(-\ln (2))} $. We can use the property $ e^{\ln (z)} = z $ to simplify the exponent. Specifically, we recognize that $ e^{-\ln(2)} = \frac{1}{2} $, and thus the expression becomes $ y(t) = 30 \left(\frac{1}{2}\right)^{t/30} $.

Step 1: Simplifying the Expression

We start with the expression for $ y(t) $: \[ y(t) = 30 e^{(t / 30)(-\ln (2))} \] Using the property $ e^{-\ln(2)} = \frac{1}{2} $, we can rewrite the expression as: \[ y(t) = 30 \left( \frac{1}{2} \right)^{t/30} \]

Step 2: Evaluating $ y(t) $ at $ t = 10 $

Next, we substitute $ t = 10 $ into the simplified expression: \[ y(10) = 30 \left( \frac{1}{2} \right)^{10/30} \] This simplifies to: \[ y(10) = 30 \left( \frac{1}{2} \right)^{1/3} \]

Step 3: Calculating the Value

Calculating $ \left( \frac{1}{2} \right)^{1/3} $ gives approximately $ 0.7937 $. Therefore: \[ y(10) \approx 30 \times 0.7937 \approx 23.8110 \]