Question

Let ^{\(H\left( t \right)\)} be the daily cost (in dollars) to heat an office building when the outside temperature is t degrees Fahrenheit.

(a) What is the meaning of ^{\(H'\left( {58} \right)\)}? What are its units?

(b) Would you expect ^{\(H'\left( {58} \right)\)} to be positive or negative? Explain.

Short answer

1. When outside temperature is \(58^\circ {\rm{F}}\), the value of ^{\(H'\left( {58} \right)\)} represents the rate at which the heating cost changes with respect to temperature. The unit of ^{\(H'\left( {58} \right)\)} is \({\rm{dollars/}}^\circ {\rm{F}}\).
2. ^{\(H'\left( {58} \right)\)}to be negative.

Step

Step-by-step solution

(a) Step 1: Meaning of \(H'\left( {58} \right)\)

When outside temperature is\(58^\circ {\rm{F}}\), the value of ^{\(H'\left( {58} \right)\)}represents the rate at which the heating cost changes with respect to temperature. The unit of^{\(H'\left( {58} \right)\)} is \({\rm{dollars/}}^\circ {\rm{F}}\).

(b) Step 2: Nature of  \(H'\left( {58} \right)\)

When the outside temperature increases, the building should require less heating; this implies that^{\(H'\left( {58} \right)\)}to be negative.