There are 20 alligators in the swamp. Each year, the number of alligators increases by 25%. There are 25 crocodiles in the swamp
. Each year, 10 new crocodiles join the swamp. Part A: Write functions to represent the number of alligators and crocodiles in the swamp throughout the years. (4 points) Part B: How many alligators are in the swamp after 4 years? How many crocodiles are in the swamp after the same number of years? (2 points) Part C: After approximately how many years is the number of alligators and crocodiles the same? Justify your answer mathematically. (4 points)
1 answer:
Answer:
Step-by-step explanation:
We have given that:
There are 20 alligators in the swamp. Each year, the number of alligators increases by 25%. There are 25 crocodiles in the swamp. Each year, 10 new crocodiles join the swamp
Part A:
According to the above mentioned statement the function for representing the number of alligators and crocodiles in the swap throughout the year is:
Alligators = y=20(1.25)^x (x represents years)
crocodiles: y=10x+25
Part B:
Number of alligators after 4 years in the swap = y=20(1.25)^x
Number of alligators after 4 years in the swap = y=20(1.25)^4
Number of alligators after 4 years in the swap=48.82
This shows that about 48 to 49 alligators will be there in the swap after 4 year
Number of crocodiles after 4 years in the swap = y= 10(4)+25 = 65
Part C:
We will use the graph method to solve this part:
After approximately 7 years, the number of animals of each type will be the same, 95.
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It is given that two vertices of square are (0,0) and (4,2).
Now the problem is that you haven't given that whether these two vertices are adjacent vertices or opposite vertices of the square.
1. By Supposing that these two are adjacent vertices of Square
The third vertex will be at (-4,2) which lies in third quadrant.
Suppose the coordinate of fourth vertex be (x,y).
Mid point of line joining (4,2) and (-4,2) is{ [4+(-4)]/2,(2+2)/2} is (0,2).
Mid point of line joining (x,y) and (0,0) is (x/2,y/2).
Since diagonals of square bisect each other,
∵ x/2=0
⇒x=0
and
y/2=2
⇒y=4
So, The Coordinate of fourth vertex is (0,4).
Now coming back to second condition if these are two opposite vertex of Square.
Let the third coordinate be (a,b).
Length of diagonal=
Now,let side of Square be A.
Then length of Diagonal of square =√2 A
⇒√2 A=2√5
⇒A =√10
As third vertex is (a,b).
Using distance formula
a² + b²=10 -------------(1)
(a-4)²+ (b-2)²=10 --------------(2)
Solving expression (1) and (2), we get
⇒a²+ b²=(a-4)² +(b-2)²
⇒2a + b =5
⇒b=5-2a
Putting the value of b in (1),we get
⇒a² +(5-2a)²=10
⇒a²+25+4a²-20a =10
⇒5a²-20a+15=0
⇒a² - 4a + 3=0
Splitting the middle term,we get
⇒(a-3)(a-1)=0
⇒a=3 ∧ a=1
we get b=5-2×1=3 and b=5-2×3=5-6=-1
So,the other vertex are (1,3) and(3,-1).
