1answer.
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
mezya [45]
3 years ago
7

50 POINTS!!! PLEASE HELP!!!

Mathematics
2 answers:
Alisiya [41]3 years ago
6 0

Let <em>t</em> be the time in years, t=0 is now.


Part A


A(t) = 30 (1.20^t)


B(t) = 45 + 3t


Part B


A(5)=  30(1.20^5) = 74.6496  \approx 75


B(5) = 45 + 3(5) = 45+14= 60


Part C


45+3t = 30(1.20^t)


Graphically for positive t these intersect around


t = 3.3 \textrm{ years}




Aleks04 [339]3 years ago
6 0

As with any involved word problem, we're gonna want to figure out what we have, what we're looking for, and find a path to get from our knowns to our unknowns.


We'll start by giving our most important numbers labels - I'll include some justifications, too:


\alpha (t) - The number of homes in neighborhood A (α) after t years

\beta (t) - The number of homes in neighborhood B (β) after t years


Why should we go straight for the functions here? Well, a function, in extremely basic terms, is a machine that takes in a number (an <em>input</em>)and produces another one (an <em>output</em>). In part B, we're given an input - 5 - and we want to see what output we get for our machines, α and β. In part C, we're told something about the outputs: we want to see when they're the same;<em> </em>what does t have to be to make \alpha (t)=\beta (t)?


Before we do any of that, we have to give our machines some rules, though, so let's lock that down first.


For \alpha (t), we're given this: <em>"There are 30 homes in Neighborhood A. Each year, the number of homes increases by 20%." </em>Here are what the first few years look like from that information:


\alpha (0)=30\\\alpha (1)=30 +30(0.2)=30(1+0.2)=30(1.2)\\a(2)=30(1.2) +30(1.2)(0.02)=30(1.2)(1+0.2)=30(1.2)(1.2)=30(1.2)^2


The pattern emerging here is <em>exponential</em> - we start with 30 homes at the beginning (\alpha (0)), and we multiply that starting value by 120% (1.2) again and again to get the number of houses for the following years. In general, we can say that


\alpha (t)=30(1.2)^t


For \beta (t), we're told: <em>"Just down the road, Neighborhood B has 45 homes. Each year, 3 new homes are built in Neighborhood B." </em>The first few years look like this:


\beta (0)=45\\\beta (1)=45 + 3\\\beta (2)=45 + 3 + 3 = 45 + 3(2)


Or, in general:


\beta (t)=45+3t


Unlike \alpha (t), \beta (t)'s growth is <em>linear</em> or <em>constant</em>. While \alpha (t) grows faster as it gets bigger, \beta (t)'s never changes. This means that, in the long term, a(t) will shoot above \beta (t). We'll see roughly when one overtakes the other in just a moment.


Now that we have our functions, we can solve parts B and C:


Part B asks how many homes neighborhoods A and B have after 5 years. Since we've already defined our rules, let's use them:


\alpha (5)=30(1.2)^5=74.6496\approx74\\\beta (5)=45+3(5)=45+15=60


After 5 years, neighborhood A has 74 homes and neighborhood B has 60. You'll notice I rounded the decimal value for \alpha (5) <em>down</em>; we don't count fractions of houses/houses under construction, so in light of that, neighborhood A has 74 <em>complete </em>houses.


For part C, we want to find out when \alpha (t)=\beta (t). We could go for an exact answer, but since the question asks for an approximation, it's fine if we just test some values and make an informed estimate.


Fortunately, we have some information that can help us. Neighborhood A has fewer homes than B at the beginning, but as we just found, A will surpass B by year 5. That means that the point where one passes the other is <em>somewhere between 0-5 years</em>. Let's test some values:


\alpha (1)=30(1.2)=36\\\beta (1)=45 + 3(1) = 48\\\\\alpha (2)=30(1.2)^2=43.2\\\beta(2)=45+3(2)=51\\\\\alpha (3) = 30(1.2)^3=51.84\\\beta (3)=45+3(3)=54\\\\\alpha (4)=30(1.2)^4=62.206\\\beta (4)=45+3(4)=57


So a good estimate for part C might be 3 1/2 years, since \alpha (t) passed \beta (t) when t was somewhere between 3 and 4.

You might be interested in
Someone help please
lys-0071 [83]

Answer:

x=4

Step-by-step explanation:

3/4x +4 = 7

3/4x +4-4 = 7-4

3/4x = 3

Multiply each side by 4/3

4/3 * 3/4x = 3 *4/3

x = 4

7 0
3 years ago
Read 2 more answers
Which group of numbers is listed from least to greatest
snow_tiger [21]

it is the second option

8 0
3 years ago
Rewrite the function by completing the square.
ddd [48]

Answer:

  f(x) = (x -6)² +14

Step-by-step explanation:

Completing the square involves writing part of the function as a perfect square trinomial.

<h3>Perfect square trinomial</h3>

The square of a binomial results in a perfect square trinomial:

  (x -h)² = x² -2hx +h²

The constant term (h²) in this trinomial is the square of half the coefficient of the linear term: h² = ((-2h)/2)².

<h3>Completing the square</h3>

One way to "complete the square" is to add and subtract the constant necessary to make a perfect square trinomial from the variable terms.

Here, we recognize the coefficient of the linear term is -12, so the necessary constant is (-12/2)² = 36. Adding and subtracting this, we have ...

  f(x) = x² -12x +36 +50 -36

Rearranging into the desired form, this is ...

  f(x) = (x -6)² +14

__

<em>Additional comment</em>

Another way to achieve the same effect is to split the given constant into two parts, one of which is the constant necessary to complete the square.

  f(x) = x² -12x +(36 +14)

  f(x) = (x² -12x +36) +14

  f(x) = (x -6)² +14

6 0
2 years ago
Enny is selling candles for $3 each at a fund-raiser. If she starts selling with $4 to help her make change, how many candles, x
ruslelena [56]

Answer:

5 candles

Step-by-step explanation:

Enny is selling candles for $3 at a fund raiser

She starts selling $4 to help her make change

Therefore the number of candles x that will be sold to get $19 can be calculated as follows

3x + 4= 19

3x = 19-4

3x= 15

x = 15/3

x= 5

Hence 5 candles must be sold to get $19

5 0
2 years ago
1/3+1/3x+8/3=-1/3(6x+11)
ivann1987 [24]

Answer:

x=-20/7 or -2.857

Step-by-step explanation:

bring like terms together in each side of the = side

then work the out

3 0
3 years ago
Other questions:
  • Please help me on number 11-12! <br><br><br><br> Thank you! :)
    5·1 answer
  • What is the diameter of a trampoline with an area of 49 square ft
    14·1 answer
  • Explain 3 different ways <br> And explain
    9·1 answer
  • Which operation will not change the value of any non zero number?
    8·1 answer
  • Factor completely 4x2-1
    11·2 answers
  • Can somebody help me with this problem
    13·2 answers
  • When I need to reduce the number by 13%, just multiply it by the number ...​
    7·1 answer
  • Is it that true ⅖-¾=¼+2/5
    11·1 answer
  • Which of the list shows number below in order from least to greates t? -5.9, -23/4, 5.78, 58%​
    9·1 answer
  • Julio watched a movie that started at 11:30 am and ended at 2:12 pm how long was the movie
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!