Math

QuestionEncuentra la temperatura máxima de la función T(x)=x2+10x2T(x) = -x^{2} + 10x - 2 para el cultivo de bacterias.

Studdy Solution

STEP 1

Assumptions1. The temperature (in Celsius) experienced by a certain bacteria culture varies according to the function (x)=x+10x(x)=-x^ +10x -. . xx represents the exposure time to heat energy sources.
3. We are asked to find the maximum temperature.

STEP 2

The given function (x)=x2+10x2(x)=-x^2 +10x -2 is a quadratic function. The maximum or minimum value of a quadratic function f(x)=ax2+bx+cf(x) = ax^2 + bx + c is given by the vertex of the parabola, which occurs at x=b2ax = -\frac{b}{2a}.

STEP 3

In our case, a=1a = -1 and b=10b =10. So, we can substitute these values into the formula to find the xx value at which the maximum temperature occurs.
x=b2ax = -\frac{b}{2a}

STEP 4

Substitute a=1a = -1 and b=10b =10 into the formula.
x=102(1)x = -\frac{10}{2(-1)}

STEP 5

Calculate the xx value.
x=102=5x = -\frac{10}{-2} =5

STEP 6

Now that we have the xx value, we can substitute this into the temperature function (x)(x) to find the maximum temperature.
(x)=x2+10x2(x) = -x^2 +10x -2

STEP 7

Substitute x=5x =5 into the function.
(5)=(5)2+10(5)2(5) = -(5)^2 +10(5) -2

STEP 8

Calculate the maximum temperature.
(5)=25+502=23(5) = -25 +50 -2 =23The maximum temperature is23 degrees Celsius.

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