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Pre-Calculus question

PreCalculus Question class=

Sagot :

The Interpret (N · T)(t) is [tex]N[T(t)] = 20t^{2}+92t+205.[/tex]

The number of bacteria after 5 hours will be 1165.

It will take for the bacteria count to reach 2221.

As per the question statement, the number of bacteria "N" in a refrigerated food is given by [N(T) = (5T² - 4T +100)], where "T" is the temperature of the food in degrees Celsius,

And when the food is removed from the refrigerator, the temperature of the food is given by [T(t) = (2t+5)].

First we are required to find out (N · T)(t), which is N[T(t)] which can be expressed as

[tex]N[T(t)] = 5[T(t)]^{2} - 4T +100\\or, N[T(t)] = 5(2t+5)^{2} -4(2t+5)+100\\or, N[T(t)] = 5(4t^{2}+25+20t)-8t-20+100\\ or, N[T(t)] = 20t^{2} +125+100t-8t+80\\ or, N[T(t)] = 20t^{2}+92t+205.[/tex]

Then, the number of bacteria after 5 hours will be N[T(5)], i.e.,

[tex]N[T(5)] = 20(5^{2})+(92*5)+205\\or, N[T(5)] = (20*25)+460+205\\or, N[T(5)] = 500+665\\or, N[T(5)]=1165[/tex]

And finally, the time required for the bacteria count to reach 2221 can be calculated by solving the quadratic equation [tex][2221 = 20t^{2}+92t+205][/tex], i.e.,

[tex][20t^{2}+92t+205 = 2221]\\or, 20t^{2}+92t+(205-2221)=0\\or,20t^{2}+92t-2016=0\\or, 5t^{2}+23t+504=0[/tex]

And the formula to calculate the solutions of a standard quadratic equation [ax² + bx +c = 0] goes as

[tex]x=\frac{-b+\sqrt{(b^{2}-4ac) } }{2a}, \frac{-b-\sqrt{(b^{2}-4ac) } }{2a}[/tex]

Comparing [ax² + bx +c = 0] with [5t² + 23t + 504=0], we get

(a = 5), (b = 23) and (c = -504), and using these values in the above mentioned formula to calculate the solutions of quadratic equations, we get, [t = 8, (-12.6)].

Since time (t) cannot be negative, thus [t ≠ (-12.6)].

Hence, (t = 8) or, the time required for the bacteria count to reach 2221 is 8 hours.

  • Quadratic Equation: In algebra, any polynomial equation whose highest degree is two and can be rearranged to the standard form of  [ax² + bx + c = 0], where (a ≠ 0), is known as a quadratic equation.

To learn about Quadratic Equations, click on the link below.

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