The rate of change of temperature with time at a point in time is given by
the derivative of the function for the temperature of the soup.
The correct responses are;
- a) H'(5) is approximately<u> -2.6 degrees Celsius per minute</u>.
- c) The equation for the line tangent is<u> y = -3.6·t + 90.8</u>
- The approximate value of C(5) is <u>72.8 °C</u>
- d) The rate of change of the temperature of the soup at t = 3 minutes is <u>-3.6 degrees Celsius per minute</u>.
Reasons:
a) From the data in the table, we have;
The approximate value of H'(5) is given by the average value of the rate of
change of the temperature with time between points, t = 3, and t = 8
Therefore;
![\displaystyle H'(5) = \mathbf{\frac{H(8) - H(3)}{8 - 3}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20H%27%285%29%20%3D%20%5Cmathbf%7B%5Cfrac%7BH%288%29%20-%20H%283%29%7D%7B8%20-%203%7D%7D)
Which gives;
![\displaystyle H'(5) = \frac{80 - 67}{8 - 3} = \mathbf{2.6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20H%27%285%29%20%3D%20%20%5Cfrac%7B80%20-%2067%7D%7B8%20-%203%7D%20%3D%20%5Cmathbf%7B2.6%7D)
Therefore, H'(5) = <u>-2.6°C per minute</u>
b) Given that the function is twice differentiable over the interval, 0 ≤ t ≤ 12, the function for the change in temperature is continuous in the interval 0 ≤ t ≤ 12
At t = 0, H(0) = 90 °C
At t = 12, H(12) = 58 °C
Therefore, there exist a temperature, of 60 °C between 90° C and 58 °C
c) The given derivative of <em>C</em> is, ![C'(t) = \mathbf{-3.6 \cdot e^{-0.05 \cdot t}}](https://tex.z-dn.net/?f=C%27%28t%29%20%3D%20%5Cmathbf%7B-3.6%20%5Ccdot%20e%5E%7B-0.05%20%5Ccdot%20t%7D%7D)
At t = 3, we have;
![The \ slope \ at \ t = 3 \ is \ C'(3) = -3.6 \cdot e^{-0.05 \times 3} \approx -3.1](https://tex.z-dn.net/?f=The%20%5C%20slope%20%5C%20at%20%5C%20t%20%3D%203%20%5C%20is%20%5C%20C%27%283%29%20%3D%20-3.6%20%5Ccdot%20e%5E%7B-0.05%20%5Ctimes%203%7D%20%5Capprox%20-3.1)
Therefore, we have;
y - 80 ≈ -3.1 × (x - 3)
The equation for the tangent is; y = -3.6 × (x - 3) + 80
y = -3.6·x + 10.8 + 80 = -3.6·x + 90.8
- The equation for the tangent is; <u>y = -3.6·x + 90.8</u>
The value of C(5) is approximately, C(5) ≈ -3.6 × 5 + 90.8 = 72.8
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d) Based on the the model above, the rate at which the temperature of the
soup is changing at t = 3 minutes is <u>-3.6 degrees per minute</u>.
Learn more about calculus and concepts here:
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