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iris [78.8K]
3 years ago
15

A roast turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the tempera

ture is 75°F.
The graph shows how the temperature of the turkey decreases and eventually approaches room temperature.

By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour.

Mathematics
1 answer:
ANEK [815]3 years ago
8 0
It is very important to look at the graph very minutely. It can be seen that at 30 minutes the slope is a little more than negative 50 and at <span>90 minutes it is a little less than negative 50. From this fact, it can be deduced that the estimated rate of </span><span>change of the temperature after an hour would be -50/60. I hope that the answer has come to your desired help.</span>
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Answer:

y(t)=2e^{3t}(2-5t)

Step-by-step explanation:

Let Y(s) be the Laplace transform Y=L{y(t)} of y(t)

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L{y'' - 6y' + 9y} = L{0} = 0

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Using the theorem of the Laplace transform for derivatives, we know that:

\large\bf L\left\{y''\right\}=s^2Y(s)-sy(0)-y'(0)\\\\L\left\{y'\right\}=sY(s)-y(0)

Replacing the initial values y(0)=4, y′(0)=2 we obtain

\large\bf L\left\{y''\right\}=s^2Y(s)-4s-2\\\\L\left\{y'\right\}=sY(s)-4

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\large\bf s^2Y(s)-4s-2-6(sY(s)-4)+9Y(s)=0

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\large\bf s^2Y(s)-4s-2-6(sY(s)-4)+9Y(s)=0\Rightarrow (s^2-6s+9)Y(s)-4s+22=0\Rightarrow\\\\\Rightarrow Y(s)=\frac{4s-22}{s^2-6s+9}

Now, we brake down the rational expression of Y(s) into partial fractions

\large\bf \frac{4s-22}{s^2-6s+9}=\frac{4s-22}{(s-3)^2}=\frac{A}{s-3}+\frac{B}{(s-3)^2}

The numerator of the addition at the right must be equal to 4s-22, so

A(s - 3) + B = 4s - 22

As - 3A + B = 4s - 22

we deduct from here  

A = 4 and -3A + B = -22, so

A = 4 and B = -22 + 12 = -10

It means that

\large\bf \frac{4s-22}{s^2-6s+9}=\frac{4}{s-3}-\frac{10}{(s-3)^2}

and

\large\bf Y(s)=\frac{4}{s-3}-\frac{10}{(s-3)^2}

By taking the inverse Laplace transform on both sides and using the linearity of the inverse:

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we know that

\large\bf L^{-1}\left\{\frac{1}{s-3}\right\}=e^{3t}

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