Find the composition function B(T(h)) Write this in simplified form.

Question

The number B of bacteria in a particular petri dish stored in a refrigerator is given by

B(T)=10T^2-60T+500.

The domain of B(T) is {T|2<=T<=14}  where T represents temperature of the petri dish in Celsius.

When we remove the petri dish from the refrigerator, the temperature of the petri dish is given by

T(h)=4h+2.

The domain of T(h) is {t|0<=h<=3} where h is the time in hours.

(a)  Find the composition function B(T(h)) Write this in simplified form.

(b) Write an interpretation of this composite function in full sentences. That is, explain what the input and output represent.

(c) Find the time when the bacteria count reaches 900 algebraically.

(2) Let f(x) have domain [-10,13] and g(x) have domain (-5,7)U(10,20]. what is the domain of f - g?. Write your solution in interval solution.

Answer 1

Given that:

B(T) = 10T² - 60T + 500

and

T(h) = 4h + 2

where:

1. B(T) is the number of bacteria when the temperature is T degrees Celsius
2. T(h) is the temperature of the petri dish after h hours

a) Finding the composition function B(T(h))

To find the composition function B(T(h)), we can plug in T(h) into B(T). We know that T(h) = 4h + 2 so we can subsitute that for T in B(T) and simplify:

B(T(h)) = 10(4h + 2)² - 60(4h + 2) + 500

= 10(4h + 2)(4h + 2) - 60(4h + 2) + 500

= 10(16h² + 8h + 8h + 4) - 240h - 120 + 500

= 10(16h² + 16h + 4) - 240h - 120 + 500

= 160h² + 160h + 40 - 240h - 120 + 500

B(T(h)) = 160h² - 80h + 420

b) Interpreting the composite function in full sentences

The composite function B(T(h)) = 160h² - 80h + 420 represents the number of bacteria in the petri dish after h hours since it was removed from the refrigerator.

1. The input is h, which is the number of hours after the dish is taken out
2. The function T(h) first tells us the temperature of the dish after h hours.
3. Then the function B(T) uses that temperature to tell us the bacteria count.

To summarize it up, the composite function B(T(h)) gives the number of bacteria in the petri dish as a function of time, where the input is time in hours after removal from the refrigerator and the output is the number of bacteria at that time.

c) Finding the time when the bacteria count reaches 900 algebraically

We use the composite function and set it equal to 900 which will gives us the time since B(T(h)) outputs the number of bacteria at time h:

160h² - 80h + 420 = 900 Move everything to one side

160h² - 80h + 420 - 900 = 0

160h² - 80h - 480 = 0 We can factor out 80

80(2h² - h - 6) = 0 Divide by 80 on both sides to get rid of it

2h² - h - 6 = 0 Factor using the ac method

In order to factor with with ac method, we know this equation is in the standard form for quadratics:

ax² + bx + c. In our equation a = 2, b = -1 and c = -6 so we multiply a and c which gives us 2(-6) = -12 so we want to find two numbers that multiply up to -12 and add up to b which is -1. Those two numbers are -4 and 3 so we can rewrite the middle term:

2h² - 4h + 3h - 6 = 0 We can using grouping to group them

(2h² - 4h) + (3h - 6) = 0 Factor each group

2h(h - 2) + 3(h - 2) = 0

(h - 2)(2h + 3) = 0 Solve each factor by setting it equal to 0

h - 2 = 0, 2h + 3 = 0

h = 2, 2h = -3

h = -3 / 2

We know that the domain of T(h) is 0 ≤ h ≤ 3, and -3/2 does not fall into that doamin so it is not allowed. Therefore, the valid answer is h = 2. We can conclude that the bacteria count reaches 900 after 2 hours.

(2) Domain of f - g

Given:

1. Domain of f(x) is [-10, 13]
2. Domain of g(x) is (-5, 7) ∪ (10, 20]

We want the domain of f - g

For f(x) - g(x) to exist, both f(x) and g(x) must be defined at the same x-values. So, we need intersection of their domains:

[-10, 13] ∩ ((-5, 7) ∪ (10, 20])

Lets first intersect with (-5, 7):

[-10, 13] ∩ (-5, 7) = (-5, 7)

because (-5, 7) is entirely inside [-10, 13].

Secondly, lets intersect with (10, 20]:

[-10, 13] ∩ (10, 20] = (10, 13]

becuse numbers greater than 10 are allowed, up to 13 is allowed by f. Also, 13 is included but 10 is not included.

Now, we can combine the pieces. So, the domain of f - g is:

Domain of f - g: (-5, 7) ∪ (10, 13]

Answer 2

B= 10t^2 -60t+500

T = 4h+2

B(T) =10(4h+2)^2-60(4h+2) +500= 160h^2 +160h + 40 -240h -120 +500

= 160h^2 -80h+420

= 20(8h^2 -4h +21)

input = time in hours, output = bacteria population

900 = 20(8h^2 -4h +21)

8h^2 -4h +21 = 45

8h^2 -4h - 24 = 0

2h^2 -h -6 = 0

(2h +3)(h-2) = 0

h = 2 hours to reach 900 population