QuestionA restaurant chef removes a beef roast from a hot oven. The temperature of the beef roast, in degrees Fahrenheit, can be modeled by the function $T(x)=78+65(0.993)^{x}$ , where $x$ is the number of minutes after the beef roast has been removed from the oven. Determine how many minutes it will take until the beef roast reaches a temperature of $125^{\circ} \mathrm{F}$ .

Studdy Solution

STEP 1

1. The temperature of the beef roast is given by the function $T(x) = 78 + 65(0.993)^x$ .
2. $x$ represents the number of minutes after the beef roast has been removed from the oven.
3. We need to find the value of $x$ when the temperature $T(x)$ is $125^\circ \mathrm{F}$ .

STEP 2

1. Set up the equation based on the given function.
2. Isolate the exponential term.
3. Solve for $x$ using logarithms.

STEP 3

Set up the equation based on the given function.
We need to find $x$ when $T(x) = 125$ .
$125 = 78 + 65(0.993)^x$

STEP 4

Isolate the exponential term.
Subtract 78 from both sides:
$125 - 78 = 65(0.993)^x$
$47 = 65(0.993)^x$
Divide both sides by 65:
$\frac{47}{65} = (0.993)^x$

STEP 5

Solve for $x$ using logarithms.
Take the natural logarithm of both sides:
$\ln\left(\frac{47}{65}\right) = \ln((0.993)^x)$
Using the logarithm power rule, $\ln(a^b) = b\ln(a)$ , we get:
$\ln\left(\frac{47}{65}\right) = x \ln(0.993)$
Solve for $x$ by dividing both sides by $\ln(0.993)$ :
$x = \frac{\ln\left(\frac{47}{65}\right)}{\ln(0.993)}$
Calculate the value of $x$ :
$x \approx \frac{-0.3646}{-0.007}$
$x \approx 52.09$
Since $x$ represents time in minutes, we round to the nearest whole number:
The beef roast will reach a temperature of $125^\circ \mathrm{F}$ in approximately $\boxed{52}$ minutes.

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