answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Yanka
2 months ago
8

Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real n

umber.
N(t)=⎧⎩⎨⎪⎪⎪⎪f(t)25t+150200+80t2+0.05tfor0≤t<6for6≤t<8fort≥8


The number of fish in a pond at time t years is modeled by the function N defined above, where f is a continuous function such that f(0)=80.


(a) Find limt→∞N(t). Explain the meaning of limt→∞N(t) in the context of the problem.

(b) Is the function N continuous at t=8 ? Justify your answer.

(c) The function N is continuous at t=6 . Is there a time t , for 0≤t≤6 , at which N(t)=250 ? Justify your answer.
Mathematics
1 answer:
AnnZ [12.3K]2 months ago
6 0

Answer:

Refer below

Step-by-step breakdown:

The function behaves as a piecewise function defined as:

N(t)=\left \{ {{25t+150} \ 0 \leq t \leq 6 \atop {\frac{200+80t}{2+0.05t} \ t \geq 8}} \right.

a)

We need to evaluate the limit of the function as t approaches infinity. This means determining the maximum number of fish present in the pond as time extends indefinitely.

We consider the second segment of the equation since t fits into that range, whereby t is infinite, clearly exceeding 8.

\lim_{t \to \infty} \frac{200+80t}{2+0.05t} \\\lim_{n \to \infty} \frac{80t}{0.05t}\\ \lim_{n \to \infty} \frac{80}{0.05} =1600

This indicates that the maximum fish population in this pond is 1600, regardless of the time.

b)

A function is considered continuous at a specific point if the limit and the function value at that point are the same.

The function value at t = 8, according to the second part of the equation, is:

\frac{200+80t}{2+0.05t}\\\frac{200+80(8)}{2+0.05(8)}\\=350

We observe that a value exists, and the limit approaches this as t nears 8.

<pthus>the function maintains continuity at t = 8

c)

We seek to determine if there exists a "time" during t from 0 to 6 when the fish count in the pond reaches 250. Substituting 250 into N(t) allows us to solve for t using the first portion of the piecewise function as shown below:

N(t)=25t+150\\250=25t+150\\25t=250-150\\25t=100\\t=4

The time is 4 years when the fish count in the pond becomes 250

</pthus>
You might be interested in
Joaquin is four more than half Marjorie’s age. Marjorie is also 32 less than three times Joaquin’s age.
babunello [11817]

Answer:

Her age is 40 years

Step-by-step explanation:

Let Jerry's age be represented as j

Let Marjorie's age be denoted as m

The first part of the question states:

j = 4 + m/2

For the second equation:

m = 3j - 32

Multiply the first equation by 2 to get:

2j = 8 + m

Leading to m = 2j - 8

Set both equations for m equal to each other:

2j - 8 = 3j - 32

By simplifying, we get:

3j - 2j = 32 - 8

Therefore, j = 24

Now substituting back:

m = 2j - 8

m = 2(24) - 8

Thus, m = 48 - 8

Final result: m = 40

5 0
1 month ago
105,159 rounded to the nearest ten thousand
Svet_ta [12734]

In this problem the number we are working with is:

105,159

By definition we note:

thousand place: a five-digit quantity greater than zero.

Moreover, the rounding rule is:

if the digit being removed is 5 or more, increase the kept digit by one.

Therefore, rounding to the nearest ten thousand yields:

105,159 = 110,000

Answer:

105,159 rounded to the nearest ten thousand is:

105,159 = 110,000

6 0
2 months ago
Read 2 more answers
Calvin deposits $400 in a savings account that accrues 5% interest compounded monthly. after c years, calvin has $658.80. makayl
lawyer [12517]
To calculate this, a specific formula will be necessary. Years = log (total/principal) / [n * log (1 + rate / n)]. Part A) For Calvin: $400 at 5% monthly results in $658.80; Time =? Monthly compounding, n = 12. Thus, Years = log(658.80/400) / [12 * log(1+ (.05/12))]. Subsequently, Years = log( 1.647) / (12 * log(1.0041666667)). Then, Years = 0.21669359917 / 12 * 0.0018058008777. Thus resulting in Years = 0.21669359917 / 0.0216696105. Ultimately, Years ≈ 9.999884362. Part B) For Makayla: $300 at 6% quarterly yields $613.04; Time=? Quarterly compounding, n = 4. Therefore, Years = log(613.04/300) / [4 * log (1 +.06/4)]. This results in Years = log(2.0434666667) / (4 * log(1.015)). Years thus equals 0.31036755784 / (4 * 0.0064660422492), resulting in Years ≈ 11.9999044949. The approximate difference is about 3 years.
3 0
15 days ago
Read 2 more answers
Five teachers were hired at Oakdale Middle School at the beginning of the 2005 school year. The performance of each teacher was
babunello [11817]
I believe the answer is Jill.
3 0
1 month ago
You have a stack of pennies without counting the pennies, how can you know if there is a odd or even number of them
Zina [12379]

Imagine having a pile of pennies and you want to determine if the count is odd or even without tallying them one by one. You can separate them into pairs by placing one coin in the left pile and one in the right.

If no coin is left unmatched, then the total number is even; if there is one coin remaining unpaired, the number is odd.

 

8 0
1 month ago
Read 2 more answers
Other questions:
  • In the diagram below BD is parallel to XY. What is the value of x
    8·2 answers
  • A store uses the expression –2p + 50 to model the number of backpacks it sells per day, where the price, p, can be anywhere from
    8·2 answers
  • On a coordinate plane, 2 lines are shown. Line P Q has points (negative 8, 2) and (4, 2). Line M N has points (8, 6) and (8, neg
    9·2 answers
  • L'età media di Aldo, Bruno, Carlo e Davide è 16 anni. Se non si tiene conto di Davide, l'età media dei tre rimanenti sale a 18.
    5·1 answer
  • Central school is hosting a “central has talent” show. They will award various prizes for the best 3 acts in the show. First pla
    8·1 answer
  • Isabella Is building a tree fort .The base of the fort is 78 inches wide by 92 inches long. What is the area of the base in squa
    9·1 answer
  • The table represents an exponential function. A 2-column table has 4 rows. The first column is labeled x with entries 1, 2, 3, 4
    10·2 answers
  • The price of fuel may increase due to demand and decrease due to overproduction. Marco is studying the change in the price of tw
    13·1 answer
  • Q. Fine Floors can install 15 square yards of carpeting in 4 hours and 30 minutes. At this rate, how
    14·2 answers
  • Sara’s little brother is getting ready to color the picture of a rainbow that Sara drew for him. To make the task easier for her
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!