A chemistry student mixes two solutions to study the properties of the resulting mixture. The temperature of the new solution is given by the function
T = 24t − 2t2 + 5, where T is the temperature in degrees Celsius and t is the time elapsed in seconds after the solutions are mixed. What is the time interval that the temperature of the solution will be at least 69°C?

Question

A chemistry student mixes two solutions to study the properties of the resulting mixture. The temperature of the new solution is given by the function 
T = 24t − 2t2 + 5, where T is the temperature in degrees Celsius and t is the time elapsed in seconds after the solutions are mixed. What is the time interval that the temperature of the solution will be at least 69°C?

welshfella · Answer

substituting for  T
69 = 24t - 2t^2 + 5

can you solve this ?

welshfella · Answer

its a quadratic equation   so we need 0  on one side :

-2x^2 + 24t - 64 = 0

to make it simpler divide through by -2 

x^2 - 12t + 32 = 0

the solutions of this equation will give you the times when T = 69

welshfella · Answer

in the original equation  we had -2t^2   so the graph will be a parabola which opens downwards  and will have a maximum value  of temperature.  The time between the 2 roots is what you require

welshfella · Answer

can you solve the equation?

welshfella · Answer

any ideas?

welshfella · Answer

x^2 - 12x + 32

this will factor 
you need 2 numbers whose product is +32 and whose sum is -12.

can you think of these 2  numbers?

welshfella · Answer

Have you studied these equations?