Q. 85 - Precalculus Enhanced with Graphing Utilities

Question

Thermostat Control Energy conservation experts estimate that homeowners can save $5\%$ to $10\%$ on winter heating bills by programming their thermostats $5$ to $10$degrees lower while sleeping. In the given graph, the temperature $T$ (in degrees Fahrenheit) of a home is given as

a function of time $t$ (in hours after midnight) over a $24$-hour period.

(a) At what temperature is the thermostat set during daytime hours? At what temperature is the thermostat set overnight?

(b) The homeowner reprograms the thermostat to $y=T\left(t\right) - 2$ 

Explain how this affects the temperature

in the house. Graph this new function.

(c) The homeowner reprograms the thermostat to $y=T(t+1)$.Explain how this affects the temperature in the house. Graph this new function.

Step-by-Step Solution

Answer

$\left(a\right)$ At $72°F$, thermostat is set during the daytime hours. At $65°F$,the thermostat is set during the overnight.

$\left(b\right)$ The graph for the function $y=T\left(t\right)-2$ is,

The temperature in the house will decrease by $2$ degrees. So, this affects the temperature in the house.

$\left(c\right)$ The graph for the function $y=T\left(t+1\right)$ is,

From the graph, it is observed that the temperature remains the same, but the daytime and over night settings change to  $1$ hour earlier.

Part ( a ) Step 1 From the given graph,

It can be seen that during the daytime hours,$8$ to $20$ hours after the thermostat is set to $72°F$.

we can see that overnight, time before and after the hours $8$ and $20$, the thermostat is set to $65°F$.

Part ( b ) Step 1 The graph of the function, y = T t - 2 is obtained by vertically shifting down the given graph by 2 units.

The temperature in the house will decrease by $2$ degrees . During the daytime hours, the temperature decreases to $70°F$ and during the night it becomes $63°F$. 

Part ( c ) Step src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHhtbG5zOndycz0iaHR0cDovL3d3dy53aXJpcy5jb20veG1sL21hdGhtbC1leHRlbnNpb24iIGhlaWdodD0iMTkiIHdpZHRoPSIxMCIgd3JzOmJhc2VsaW5lPSIxNSI+PCEtLU1hdGhNTDogPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtbj4xPC9tbj48L21hdGg+LS0+PGRlZnM+PHN0eWxlIHR5cGU9InRleHQvY3NzIi8+PC9kZWZzPjx0ZXh0IGZvbnQtZmFtaWx5PSJBcmlhbCIgZm9udC1zaXplPSIxNiIgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iNC41IiB5PSIxNSI+MTwvdGV4dD48L3N2Zz4=" role="math" localid="1645946120107" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/13394f77-3551-42fc-be85-1fe4929b7261.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220227%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220227T074209Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=19dbefc5aab15c1b46530c95ce5d9044233abdfd1a46c7e64df8fc68a2d15a81" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/13394f77-3551-42fc-be85-1fe4929b7261.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220227%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220227T074057Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=804047b7bf866fc13286daa459ab961c3064554c6ca8e8b4bcce2a55a629763d" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/13394f77-3551-42fc-be85-1fe4929b7261.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220227%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220227T073606Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=61e16f478242c867dc61e2e4ea36ea5fa82277bc1e290a08c371122f02a81362" 1 The function y = T ( t + 1 ) is graphed by shifting the graph to the left by 1 unit.

we can see that , the temperature remains the same , but the daytime and overnight settings change to $1$ hour earlier. That is the thermostat starts warming up at instead of and cools down  from instead of.