Question

Certainly! Let's consider a different scenario: = The temperature of a hot object is given by the function T(t) = 1000-e-0.02t +300, where T(t) is the temperature at time t in minutes, and the temperature is measured in degrees Celsius. At what time will the temperature be half of its initial value?​

Answer 1

The temperature is half of its initial value at approximately
`\(t \approx 115.52\)` minutes, as determined by solving the equation
`\(T(t) = (1)/(2) \cdot T(0)\)`.

Certainly! To find the time at which the temperature is half of its initial value, we set (T(t)) equal to half of its initial value and solve for (t).

The initial temperature ((T(0))) is given by
`\(1000 - e^(-0.02 \cdot 0) + 300 = 1000 + 300 = 1300\)`.

Now, we set
`\(T(t) = (1)/(2) \cdot T(0)\)`:

`\[1000 - e^(-0.02t) + 300 = (1)/(2) \cdot 1300.\]`

First, subtract 300 from both sides:

`\[1000 - e^(-0.02t) = (1)/(2) \cdot 1000.\]`

Now, subtract 1000 from both sides:

`\[-e^(-0.02t) = -500.\]`

Multiply both sides by -1 to simplify:

`\[e^(-0.02t) = 500.\]`

To solve for (t), take the natural logarithm (ln) of both sides:

`\[-0.02t = \ln(500).\]`

Now, isolate (t) by dividing both sides by -0.02:

`\[t = (\ln(500))/(-0.02).\]`

Use a calculator to find the numerical value of (t):

`\[t \approx (\ln(500))/(-0.02) \approx 115.52 \text{ minutes}.\]`

Therefore, the temperature will be half of its initial value approximately at
`\(t \approx 115.52\)` minutes.