Answered

Get the most out of your questions with the extensive resources available on IDNLearn.com. Get the information you need from our community of experts who provide accurate and comprehensive answers to all your questions.

Select the correct answer.

The number of bacteria [tex](b)[/tex] in a batch of refrigerated food is given by the function [tex]b(t)=20 t^2-70 t+300[/tex], where [tex]t[/tex] is the temperature of the food in degrees Fahrenheit. The function [tex]t(h)=2 h+3[/tex] gives the temperature of the food [tex]h[/tex] hours after it is removed from refrigeration. Which expression correctly represents the number of bacteria [tex](b)[/tex] in the food as a function of the number of hours [tex](h)[/tex] the food is unrefrigerated?

A. [tex]80 h^2 + 100 h + 690[/tex]
B. [tex]80 h^2 + 100 h + 270[/tex]
C. [tex]40 h^2 + 140 h + 570[/tex]
D. [tex]40 h^2 - 140 h + 603[/tex]

Sagot :

GinnyAnsweridn
Alright, let's go through the problem step-by-step to find the correct expression for the number of bacteria in terms of the number of hours [tex]\( h \)[/tex] the food has been unrefrigerated.

We start with the following:
1. The number of bacteria [tex]\( b \)[/tex] is given by:
[tex]\[ b(t) = 20t^2 - 70t + 300 \][/tex]
where [tex]\( t \)[/tex] is the temperature in degrees Fahrenheit.
2. The temperature in terms of hours [tex]\( h \)[/tex] is given by:
[tex]\[ t(h) = 2h + 3 \][/tex]

To find the number of bacteria [tex]\( b \)[/tex] as a function of [tex]\( h \)[/tex], we need to substitute the expression for [tex]\( t \)[/tex] into the equation for [tex]\( b(t) \)[/tex].

Let's do this substitution step-by-step:

- Substitute [tex]\( t = 2h + 3 \)[/tex] into [tex]\( b(t) \)[/tex]:
[tex]\[ b(t(h)) = 20(2h + 3)^2 - 70(2h + 3) + 300 \][/tex]

Next, we need to expand the expression [tex]\( (2h + 3)^2 \)[/tex]:

- Expand [tex]\( (2h + 3)^2 \)[/tex] using the algebraic identity [tex]\( (a+b)^2 = a^2 + 2ab + b^2 \)[/tex]:
[tex]\[ (2h + 3)^2 = (2h)^2 + 2 \cdot 2h \cdot 3 + 3^2 \][/tex]
[tex]\[ (2h + 3)^2 = 4h^2 + 12h + 9 \][/tex]

Now, substitute [tex]\( 4h^2 + 12h + 9 \)[/tex] back into the bacteria function:
[tex]\[ b(2h + 3) = 20(4h^2 + 12h + 9) - 70(2h + 3) + 300 \][/tex]

Expand and simplify each term:

1. Distribute 20:
[tex]\[ 20(4h^2 + 12h + 9) = 80h^2 + 240h + 180 \][/tex]

2. Distribute -70:
[tex]\[ -70(2h + 3) = -140h - 210 \][/tex]

3. Combine all terms together, including the constant 300:
[tex]\[ b(2h + 3) = 80h^2 + 240h + 180 - 140h - 210 + 300 \][/tex]

Now, combine like terms:
1. Combine the [tex]\( h^2 \)[/tex] terms:
[tex]\[ 80h^2 \][/tex]

2. Combine the [tex]\( h \)[/tex] terms:
[tex]\[ 240h - 140h = 100h \][/tex]

3. Combine the constant terms:
[tex]\[ 180 - 210 + 300 = 270 \][/tex]

So, the expression representing the number of bacteria [tex]\( b \)[/tex] in the food as a function of the number of hours [tex]\( h \)[/tex] the food is unrefrigerated is:
[tex]\[ b(h) = 80h^2 + 100h + 270 \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{80h^2 + 100h + 270} \][/tex]
which corresponds to option B.
Your participation is crucial to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. Find clear answers at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.