Question

When is the body temperature for $ y = - 8{x^2} + 64x - 120 $ at its peak? What is its value then?
Options
(A) 4hrs, \[{106.4^\circ }{\text{F}}\]
(B) $ 3{\text{hrs}},{102.4^\circ }{\text{F}} $
(C) $ 2{\text{hrs}},{104.4^\circ }{\text{F}} $
(D) \[1hrs,{98^\circ }{\text{F}}\]

Answer 1

Hint: Temperature is a physical quantity that describes how hot or cold something is. When a body comes into contact with another that is cooler or cooler, it is the origin of thermal energy, which is inherent in all matter and is the cause of the occurrence of heat, a flow of energy.

Complete answer:
A differential equation is a mathematical equation that connects one or more functions and their derivatives. In implementations, functions are used to describe physical quantities, derivatives are used to represent their rates of change, and the differential equation is used to describe a relationship between them. Differential equations play an important role in many fields, including chemistry, physics, economics, and biology, since such relationships are general.
The analysis of differential equations primarily entails looking at their solutions (the set of functions that satisfy each equation) as well as their properties. Only the most basic differential equations can be solved using explicit formulas; moreover, certain properties of solutions to a differential equation can be calculated without calculating them precisely.
Now given
\[y = - 8{x^2} + 64x - 120,{\text{ }}x > 0\]
Here
The y component is temperature in fahrenheit and x component is time in hours
Upon differentiating
 $ \dfrac{{dy}}{{dt}} = - 16x + 64 = 0 $
Solving for x we get
 $ \Rightarrow x = \dfrac{{64}}{{16}} = 4 $
Hence the maximum can be referred to as 4 hours.
Let the normal body temperature be $ {T_o} $
 $ {T_0} = {98.4^\circ }{\text{F}} $ $ $
When x = 4
 $ y = - 8{(4)^2} + 64(4) - 120 $
 $ y = - 128 + 256 - 120 = {8^\circ }F $
The peak temperature is the temperature at which the heat flow signal deviates the most from the simulated baseline. At this temperature, the sample material in the DSC melts totally in the case of pure, homogeneous compounds.
Peak temperature is considered as
 $ T = {T_0} + y $
 $ T = 98.4 + 8 = {106.4^\circ }{\text{F}} $
Hence option A is correct.

Note:
If there isn't a closed-form expression for the solution, it's always possible to calculate it numerically using machines. Many computational methods have been developed to evaluate solutions with a given degree of precision, whereas the theory of dynamical systems emphasises qualitative study of systems represented by differential equations.