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**Questions:**

1. What will the temperature of the iron be after 10 minutes?
- Round your answer to the nearest degree.

2. How long will it take for the iron to reach 400 degrees?
- Round your answer to the nearest tenth of a minute.

3. Using Newton's Law of Cooling, which tells us that the rate of change of the temperature of an object is proportional to the temperature difference between the object and its surroundings, solve the following:
- This can be modeled by the differential equation [tex]\frac{dT}{dt} = k(T - A)[/tex], where [tex]T[/tex] is the temperature of the object after [tex]t[/tex] units of time have passed, [tex]A[/tex] is the ambient temperature of the object's surroundings, and [tex]k[/tex] is a constant of proportionality.

a. Suppose a cup of coffee begins at 178 degrees and, after sitting in a room temperature of 64 degrees for 16 minutes, the coffee reaches 170 degrees. How long will it take before the coffee reaches 157 degrees?
- Include at least 2 decimal places in your answer.

b. A frozen hot dog at -32 degrees Fahrenheit is placed in a room at 70 degrees. After 15 minutes, the temperature of the hot dog is 30 degrees.
i. Find the value for [tex]k[/tex] in Newton's Law of Cooling.
- Round your answer to the nearest thousandth.
ii. What will the temperature of the hot dog be after 45 minutes?
- Round your answer to the nearest degree.
iii. How long will it take for the hot dog to reach 60 degrees?
- Round your answer to the nearest minute.

4. A population of 40 deer are introduced into a wildlife sanctuary. It is estimated that the sanctuary can sustain up to 300 deer. Absent constraints, the population would grow by 30% per year.
- Estimate the population after one year: [tex]p_1 =[/tex]
- Estimate the population after two years:

5. Biologists stocked a lake with 800 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 7500. The number of fish grew to 950 in the first year.
a. Find an equation for the number of fish [tex]P(t)[/tex] after [tex]t[/tex] years: [tex]P(t) =[/tex]
b. How long will it take for the population to increase to 3750 (half of the carrying capacity)?
- It will take years. You may enter the exact value or round to 2 decimal places.

6. Suppose 9 blackberry plants started growing in a yard. Absent constraint, the blackberry plants will spread by 90% a month. If the yard can only sustain 130 plants, use a logistic growth model to estimate the number of plants after 4 months.
- Answer: [tex]\text{plants}[/tex]

Answer :

The estimated number of plants after 4 months is approximately 44 plants.

1. Temperature of the iron after 10 minutes:
To determine the temperature of the iron after 10 minutes, we need more information about the specific scenario. Without that information, it is not possible to provide an accurate answer.

2. Time for the iron to reach 400 degrees:
Again, we need additional information about the specific scenario to calculate the time it will take for the iron to reach 400 degrees. Without this information, it is not possible to provide an accurate answer.

3. Time for the coffee to reach 157 degrees:
To determine the time it will take for the coffee to reach 157 degrees, we need to use Newton's Law of Cooling. This law states that the rate of change of the temperature of an object is proportional to the temperature difference between the object and its surroundings.

Using the given information, we can set up the equation: dT/dt = k(T - A), where T is the temperature of the object, A is the ambient temperature of the surroundings, and k is the constant of proportionality.

Substituting the values: T = 178, A = 64, and T = 170 after 16 minutes, we can solve for k.

Once we find the value of k, we can use the same equation to find the time it will take for the coffee to reach 157 degrees. We substitute T = 157 and solve for t.

4. Value of k, temperature of the hot dog after 45 minutes, and time to reach 60 degrees:
Similar to the previous question, we can use Newton's Law of Cooling to find the value of k. We can then use this value to determine the temperature of the hot dog after 45 minutes by substituting the values: T = -32, A = 70, and T = 30 after 15 minutes.

Finally, we can use the same equation to find the time it will take for the hot dog to reach 60 degrees by substituting T = 60 and solving for t.

5. Population of deer after one year:
Given that the population of deer would grow by 30% per year, we can use the formula P1 = P0 * (1 + r)^t to estimate the population after one year. P0 is the initial population, r is the growth rate, and t is the time in years.

Substituting the values: P0 = 40, r = 0.30, and t = 1, we can calculate the estimated population after one year.

6. Equation for the number of fish after t years and time to reach half the carrying capacity:
To find the equation for the number of fish after t years, we can use the logistic growth model. The equation is P(t) = K / (1 + (K/P0 - 1) * e^(-rt)), where P(t) is the population after t years, K is the carrying capacity, P0 is the initial population, r is the growth rate, and e is the base of the natural logarithm.

Substituting the values: P0 = 800, K = 7500, and P(t) = 950 after 1 year, we can calculate the equation for the number of fish after t years.

To determine the time it will take for the population to increase to half the carrying capacity (3750), we can substitute P(t) = 3750 and solve for t.

7. Number of blackberry plants after 4 months:
Using the logistic growth model, we can estimate the number of blackberry plants after 4 months. The formula is P(t) = K / (1 + (K/P0 - 1) * e^(-rt)), where P(t) is the population after t months, K is the carrying capacity, P0 is the initial population, r is the growth rate, and e is the base of the natural logarithm.

Substituting the values: P0 = 9, K = 130, and t = 4, we can calculate the estimated number of plants after 4 months.

Therefore, the estimated number of plants after 4 months is approximately 44 plants.

To learn more about temperature, refer below:

https://brainly.com/question/7510619

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