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Answer :
Let's solve each part of the question step by step.
### a. What was the temperature of the oven when it was turned on?
To find the temperature of the oven when it was turned on, we need to find the value of [tex]\( T \)[/tex] when [tex]\( m = 0 \)[/tex]. The formula given is:
[tex]\[ T = 400 - 350 \cdot (32)^{-0.1m} \][/tex]
Plugging in [tex]\( m = 0 \)[/tex]:
[tex]\[ T = 400 - 350 \cdot (32)^{-0.1 \times 0} \][/tex]
Since any number raised to the power of 0 is 1:
[tex]\[ T = 400 - 350 \cdot 1 \][/tex]
[tex]\[ T = 400 - 350 \][/tex]
[tex]\[ T = 50 \][/tex]
So, the temperature of the oven when it was turned on was [tex]\( 50^\circ \text{F} \)[/tex].
### b. How many minutes will it take for the oven's temperature to reach [tex]\( 300^\circ F \)[/tex]?
We need to determine [tex]\( m \)[/tex] when [tex]\( T = 300 \)[/tex]. Use the equation:
[tex]\[ 300 = 400 - 350 \cdot (32)^{-0.1m} \][/tex]
Rearrange to solve for [tex]\( m \)[/tex]:
1. Subtract 400 from both sides:
[tex]\[ -100 = -350 \cdot (32)^{-0.1m} \][/tex]
2. Divide both sides by -350:
[tex]\[ \frac{100}{350} = (32)^{-0.1m} \][/tex]
[tex]\[ \frac{2}{7} = (32)^{-0.1m} \][/tex]
3. Take the logarithm of both sides to solve for [tex]\( m \)[/tex]:
[tex]\[ -0.1m \cdot \log(32) = \log\left(\frac{2}{7}\right) \][/tex]
4. Solve for [tex]\( m \)[/tex]:
[tex]\[ m = \frac{\log\left(\frac{2}{7}\right)}{-0.1 \cdot \log(32)} \][/tex]
By solving the equation above, it is determined that:
- The number of minutes [tex]\( m \)[/tex] it will take for the oven's temperature to reach [tex]\( 300^\circ F \)[/tex] is approximately [tex]\( 3.6147 \)[/tex].
Rounding to the nearest minute gives:
[tex]\[ m \approx 4 \text{ minutes} \][/tex]
Therefore, it will take about 4 minutes for the oven to reach [tex]\( 300^\circ F \)[/tex].
### a. What was the temperature of the oven when it was turned on?
To find the temperature of the oven when it was turned on, we need to find the value of [tex]\( T \)[/tex] when [tex]\( m = 0 \)[/tex]. The formula given is:
[tex]\[ T = 400 - 350 \cdot (32)^{-0.1m} \][/tex]
Plugging in [tex]\( m = 0 \)[/tex]:
[tex]\[ T = 400 - 350 \cdot (32)^{-0.1 \times 0} \][/tex]
Since any number raised to the power of 0 is 1:
[tex]\[ T = 400 - 350 \cdot 1 \][/tex]
[tex]\[ T = 400 - 350 \][/tex]
[tex]\[ T = 50 \][/tex]
So, the temperature of the oven when it was turned on was [tex]\( 50^\circ \text{F} \)[/tex].
### b. How many minutes will it take for the oven's temperature to reach [tex]\( 300^\circ F \)[/tex]?
We need to determine [tex]\( m \)[/tex] when [tex]\( T = 300 \)[/tex]. Use the equation:
[tex]\[ 300 = 400 - 350 \cdot (32)^{-0.1m} \][/tex]
Rearrange to solve for [tex]\( m \)[/tex]:
1. Subtract 400 from both sides:
[tex]\[ -100 = -350 \cdot (32)^{-0.1m} \][/tex]
2. Divide both sides by -350:
[tex]\[ \frac{100}{350} = (32)^{-0.1m} \][/tex]
[tex]\[ \frac{2}{7} = (32)^{-0.1m} \][/tex]
3. Take the logarithm of both sides to solve for [tex]\( m \)[/tex]:
[tex]\[ -0.1m \cdot \log(32) = \log\left(\frac{2}{7}\right) \][/tex]
4. Solve for [tex]\( m \)[/tex]:
[tex]\[ m = \frac{\log\left(\frac{2}{7}\right)}{-0.1 \cdot \log(32)} \][/tex]
By solving the equation above, it is determined that:
- The number of minutes [tex]\( m \)[/tex] it will take for the oven's temperature to reach [tex]\( 300^\circ F \)[/tex] is approximately [tex]\( 3.6147 \)[/tex].
Rounding to the nearest minute gives:
[tex]\[ m \approx 4 \text{ minutes} \][/tex]
Therefore, it will take about 4 minutes for the oven to reach [tex]\( 300^\circ F \)[/tex].
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