# Find the upper and lower estimates for the temperature? The given values are attached as a picture! Thanks.

## Feb 11, 2018

lower: 65.8°C
upper: 73°C
(note: the stone's temperature is between these 2 temperature but can't actually be either of them)

#### Explanation:

the graphs below might help the explanation make more sense.

the highest temperature of the stone after 4 minutes is if the stone continues increasing at 15°C/min (the initial rate) (see first graph). since the temperature graph is concave down, the maximum temperature of the stone after 4 minutes will be an infinitely small amount less than this linear approximation:
$y - 13 = 15 \left(x - 0\right) \mathmr{and} y = 15 x + 13$ (this is the higher of the 2 dashed lines on the graph)

when x = 4, y = 15(4)+13 = 73, the maximum.

the lowest temperature of the stone after 4 minutes is if the stone temperature increases at a rate just enough to reach 79°C by 5 mins (see second graph). to find the linear approximation, first find the slope of the line containing (0,13) and (5,79)

slope = 66/5

linear approximation:
$y - 13 = \frac{66}{5} \left(x - 0\right) , y = \frac{66}{5} x + 13$
when x = 4, y = 66/5(4)+13 = 65.8, the minimum

max: min: 