a can of soda at 80 F is placed in a refrigerator that maintains a constant temperature of 37 F the temperature T of the soda t minutes after it is placed in the refrigerator is given by
T(t)=37+43e^-0.058t
find temp to the nearest degree of the soda after 7 minutes
when to the nearest minute will the temp of the soda be 43 F
T(7) = 37+43e^-0.058(7) = 65.65
43 = 37+43e^-0.058t
6/43 = .1395 = e^-0.058t
ln .1395 = -.058t
−1.9696 = -.058t
t = 33.96
To find the temperature of the soda after 7 minutes, we need to substitute the value of t = 7 into the equation T(t) = 37 + 43e^(-0.058t).
1. Start by substituting t = 7 into the equation:
T(t) = 37 + 43e^(-0.058 * 7)
2. Calculate the exponential term:
e^(-0.058 * 7) ≈ 0.4275
3. Substitute the value of e^(-0.058 * 7) into the equation:
T(t) = 37 + 43 * 0.4275
4. Perform the multiplication:
T(t) ≈ 37 + 18.3675
5. Add the numbers:
T(t) ≈ 55.3675
Therefore, to the nearest degree, the temperature of the soda after 7 minutes will be approximately 55°F.
Now, let's find when the temperature of the soda will be 43°F.
1. Start with the equation T(t) = 37 + 43e^(-0.058t) and set it equal to 43:
43 = 37 + 43e^(-0.058t)
2. Subtract 37 from both sides of the equation:
43 - 37 = 43e^(-0.058t)
3. Simplify the left side of the equation:
6 = 43e^(-0.058t)
4. Divide both sides of the equation by 43:
6/43 = e^(-0.058t)
5. Take the natural logarithm (ln) of both sides of the equation:
ln(6/43) = -0.058t
6. Divide both sides of the equation by -0.058 to isolate t:
t = ln(6/43) / -0.058
7. Plug the value into a calculator to find the approximate value of t:
t ≈ 25 minutes
Therefore, to the nearest minute, the temperature of the soda will be approximately 43°F after 25 minutes.