Question

Between 0°C and 30°C, the volume V ( in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula V = 999.87 − 0.06426T + 0.0085043T2 − 0.0000679T3 Find the temperature at which water has its maximum density. (Round your answer to four decimal places.)

Answer 1

Answer:

temperature is  3.9665

Explanation:

volume is given in the form of equation with respect to temperature

$V = 999.87 − 0.06426T + 0.0085043T^2 − 0.0000679T^3$

we know that, Density is max when volume is minimum.

therefore find the derivatives of the above equation:

V' = -0.0002037 T^2 + 0.0170086 T - 0.06426

for critical point set above {dv/dt = 0} equation equal to zero

-0.0002037 T^2 + 0.0170086 T - 0.06426 = 0

by using below formula find the root of above equation

$T = \frac{ -b \pm  \sqrt{ b^2 - 4ac}}{2(a)}$

$T = \frac { - 0.0170086 \pm \sqrt { 0.0170086 ^ { 2 } - 4 ( - 0.0002037 ) ( - 0.06426 ) } } { 2 ( - 0.0002037 ) }$

T≈3.9665, 79.5318

Don't consider 79.5318, as this value lie  outside the given range i.e. 0°C and 30°C.

Hence, temperature is  3.9665

Answer 2

The water has its maximum density at a temperature of approximately 3.967 °C.

Volume ($V$), in cubic centimeters, and density ($\rho$), in kilograms per cubic centimeters, are inversely proportional. Maximum density is found when volume is a minimum, which is found by first and second derivative tests:

Original function

$V = 999.87 - 0.06426\cdot T + 0.0085043\cdot T^{2} - 0.0000679\cdot T^{3}$ (1)

First derivative test

$-0.06426 + 0.0170086\cdot T - 2.037\cdot 10^{-4}\cdot T^{2} = 0$

All the roots are described below:

$T_{1} \approx 79.532\,^{\circ}C$, $T_{2}\approx 3.967\,^{\circ}C$

The only solution that is mathematically reasonable is:

$T\approx 3.967\,^{\circ}C$

Second derivative test

$V'' = 0.0170086 -4.074\cdot 10^{-4}\cdot T$

$V'' = 0.0170086 - 4.074\times 10^{-4}\cdot (3.967\,^{\circ}C)$

$V'' = 0.0153$

The temperature given leads to a maximum density.

The water has its maximum density at a temperature of approximately 3.967 °C.

We kindly invite to check this question on maxima and minima: https://brainly.com/question/12870574