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Chapter 7: Problem 74

The temperature, in degrees Fahrenheit, of a dessert placed in a freezer for \(t\) hours is modeled by $$ \frac{t+30}{t^{2}+4 t+1}-\frac{t-50}{t^{2}+4 t+1} $$ a. Express the temperature as a single rational expression. b. Use your rational expression from part (a) to find the temperature of the dessert, to the nearest hundredth of a degree, after 1 hour and after 2 hours.

Short Answer

Expert verified
The temperature, in degrees Fahrenheit, of a dessert placed in a freezer for \(t\) hours is given by \(\frac{80}{t^{2}+4 t+1}\). After 1 hour, the temperature of the dessert will be 13.33 degrees Fahrenheit, and after 2 hours, it will be 5.71 degrees Fahrenheit, each to the nearest hundredth of a degree.

Step by step solution

01

Simplify the Expression

In the expression given, \(\frac{t+30}{t^{2}+4 t+1}-\frac{t-50}{t^{2}+4 t+1}\), two fractions with the same denominators are subtracted. To express that as a single rational expression, these fractions can be combined by subtracting the numerators. You get: \(\frac{(t+30)-(t-50)}{t^{2}+4 t+1}\). This simplifies further to \(\frac{80}{t^{2}+4 t+1}\).
02

Substitute t = 1

With expression simplified as \(\frac{80}{t^{2}+4 t+1}\), calculate the temperature after 1 hour by substituting \(t=1\). The result is \(\frac{80}{1^{2}+4(1)+1} = \frac{80}{6} = 13.33\) degrees Fahrenheit, to the nearest hundredth of a degree.
03

Substitute t = 2

Next, calculate the temperature after 2 hours by substituting \(t=2\) into the simplified equation. The result is \(\frac{80}{2^{2}+4(2)+1} = \frac{80}{14} = 5.71\) degrees Fahrenheit, to the nearest hundredth of a degree.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. a. If \(y=\frac{k}{x},\) find the value of \(k\) using \(x=8\) and \(y=12\) b. Substitute the value for \(k\) into \(y=\frac{k}{x}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=3\)

Describe two similarities between the following problems: $$ \frac{3}{8}+\frac{1}{8} \text { and } \frac{x}{x^{2}-1}+\frac{1}{x^{2}-1} $$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3}{x-2}+\frac{4}{x+3}$$

The temperature, in degrees Fahrenheit, of a dessert placed in a freezer for \(t\) hours is modeled by $$ \frac{t+30}{t^{2}+4 t+1}-\frac{t-50}{t^{2}+4 t+1} $$ a. Express the temperature as a single rational expression. b. Use your rational expression from part (a) to find the temperature of the dessert, to the nearest hundredth of a degree, after 1 hour and after 2 hours.

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x}{x-y}+\frac{y}{y-x}$$
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