Math (Quadractic function)

posted by CLINT

Hi can you help me find the solution for this one? I have the answers at the end but I want to study how did it end with that. Thanks :)

The number N of bacteria in a refrigerated food is given by

N(T) = 10T^2 - 20T + 600, 1 ≤ T ≤ 20

where T is the temperature of food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by

T(t) = 3t + 2, 0 ≤ t ≤ 6

where t is time in hours.

a. Find the composition N(T(t)).

b. Find the bacteria count after 0.5 hour.

c. Find the time when the bacteria count reaches 1500.

---

Answers:

a. N(T(t))= 30(3t^2+2t+20)

b. About 653 bacteria

c. 2.846 hour

  1. Bosnian

    a.

    N ( T ) = 10 T ^ 2 - 20 T + 600

    T ( t ) = 3 t + 2


    N ( T ( t ) ) = 10 * ( 3 t + 2 ) ^ 2 - 20 * ( 3 t + 2 ) + 600 =

    10 * [ ( 3 t ) ^ 2 + 2 * 3 t * 2 + 2 ^ 2 ] - 20 * 3 t - 20 * 2 + 600 =

    10 * ( 9 t ^ 2 + 12 t + 4 ) - 60 t - 40 + 600 =

    90 t ^ 2 + 120 t + 40 - 60 t - 40 + 600 =

    90 t ^ 2 + 60 t + 600 =

    30 * 3 t ^ 2 + 30 * 2 t + 30 * 20 =

    30 ( 3 t ^ 2 + 2 t + 20 )


    b.

    t = 0.5 h


    N = 30 ( 3 t ^ 2 + 2 t + 20 )

    N ( T ( 0.5 h ) ) = 30 ( 3 * 0.5 ^ 2 + 2 * 0.5 + 20 ) =

    30 ( 3 * 0.25 + 1 + 20 ) =

    30 ( 0.75 + 1 + 20 ) =

    30 * 21.75 = 652.5

    approx. 653


    c.

    30 ( 3 t ^ 2 + 2 t + 20 ) = 1500 Divide both sides by 30

    3 t ^ 2 + 2 t + 20 = 50 Subtract 50 to both sides

    3 t ^ 2 + 2 t + 20 - 50 = 50 - 50

    3 t ^ 2 + 2 t - 30 = 0

    The solutions are :

    t = ( sqrt 91 - 1 ) / 3 = 2.846464 h

    and

    t = ( - 1 - sqrt 91 ) / 3 = - 3.51313 h

    Time can't be negative so ;

    t = 2.846464 h


    P.S.

    If you don't know how to solve equation

    3 t ^ 2 + 2 t - 30 = 0

    In google type :

    quadratic equation online

    When you see list of results click on :

    w w w . w e b g r a p h i n g . c o m / q u a d r a t i c e q u a t i o n _ q u a d r a t i

    When page be open in rectangle type :

    3 t ^ 2 + 2 t - 30 = 0

    then click option :

    solve it!

    You will see solution step - by - step.

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