Questions: Graphs and Functions Finding, evaluating, and interpreting an inverse function for a given linea... 3,5 Tara Scientists have found a relationship between the temperature and the height above a distant planet's surface. T(h), given below, is the temperature in Celsius Español at a height of h kilometers above the planet's surface. The relationship is as follows. T(h) = 30.5 - 2.5h Complete the following statements. Let T^-1 be the inverse function of T. Take x to be an output of the function T. That is, x = T(h) and h = T^-1(x). (a) Which statement best describes T^-1(x) ? The temperature (in degrees Celsius) at a height of x kilometers. The ratio of the temperature (in degrees Celsius) to the number of kilometers, x. The reciprocal of the temperature (in degrees Celsius) at a height of x kilometers. The height above the surface (in kilometers) when the temperature is x degrees Celsius. (b) T^-1(x) = (x - 30.5) / 2.5 (c) T^-1(15) = 6.2

Graphs and Functions Finding, evaluating, and interpreting an inverse function for a given linea... 3,5 Tara Scientists have found a relationship between the temperature and the height above a distant planet's surface. T(h), given below, is the temperature in Celsius Español at a height of h kilometers above the planet's surface. The relationship is as follows. T(h) = 30.5 - 2.5h Complete the following statements. Let T^-1 be the inverse function of T. Take x to be an output of the function T. That is, x = T(h) and h = T^-1(x). (a) Which statement best describes T^-1(x) ? The temperature (in degrees Celsius) at a height of x kilometers. The ratio of the temperature (in degrees Celsius) to the number of kilometers, x. The reciprocal of the temperature (in degrees Celsius) at a height of x kilometers. The height above the surface (in kilometers) when the temperature is x degrees Celsius. (b) T^-1(x) = (x - 30.5) / 2.5 (c) T^-1(15) = 6.2
Transcript text: Graphs and Functions Finding, evaluating, and interpreting an inverse function for a given linea... 3,5 Tara Scientists have found a relationship between the temperature and the height above a distant planet's surface. $T(h)$, given below, is the temperature in Celsius Español at a height of $h$ kilometers above the planet's surface. The relationship is as follows. \[ T(h)=30.5-2.5 h \] Complete the following statements. Let $T^{-1}$ be the inverse function of $T$. Take $x$ to be an output of the function $T$. That is, $x=T(h)$ and $h=T^{-1}(x)$. (a) Which statement best describes $T^{-1}(x)$ ? The temperature (in degrees Celsius) at a height of $x$ kilometers. The ratio of the temperature (in degrees Celsius) to the number of kilometers, $x$. The reciprocal of the temperature (in degrees Celsius) at a height of $x$ kilometers. The height above the surface (in kilometers) when the temperature is $x$ degrees Celsius. (b) $T^{-1}(x)=\frac{x-30.5}{2.5}$ (c) $T^{-1}(15)=6.2$
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Solution

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Solution Steps

Step 1: Given the linear function

Given the linear temperature-height relationship on a distant planet is described by the equation $T(h) = a - b \cdot h$, where $T(h)$ is the temperature in Celsius at a height $h$ kilometers above the planet's surface, $a$ is the base temperature at the surface level, and $b$ is the rate of temperature decrease per kilometer.

Step 2: Solve for $h$

To find the inverse function $T^{-1}(x)$, we solve the equation $x = a - b \cdot h$ for $h$. Rearranging the equation gives $h = \frac{a - x}{b}$. That is $h = \frac15.52.5$.

Step 3: Calculate the inverse function $T^{-1}(x)$

Substituting $a = 30.5$, $b = 2.5$, and $x = 15$ into the equation $h = \frac{a - x}{b}$ gives $h = \frac{30.5 - 15}{2.5} = 6.2$.

Final Answer:

The height above the planet's surface when the temperature is 15 degrees Celsius is approximately 6.2 kilometers.

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