Given that a species of beetles grows 32% every year.
So growth rate is given by
r=32%= 0.32
Given that 100 beetles are released into a field.
So that means initial number of beetles P=100
Now we have to find about how many beetles will there be in 10 years.
To find that we need to setup growth formula which is given by
where A is number of beetles at any year n.
Plug the given values into above formula we get:
now plug n=10 years
Hence answer is approx 1606 beetles will be there after 10 years.
To find answer for 20 years plug n=20 years
Hence answer is approx 25791 beetles will be there after 20 years.
Now we have to find time for 100000 beetles so plug A=100000
24.8810001465=n
Hence answer is approx 25 years.
Using 2 + .025x, the 2 represents the fixed rental rate per book, and 0.25 is the late fee per day
Answer:
The coefficients are therefore
1,1
1,2,1
1,3,3,1
1,4,6,4,1
1,5,10,10,5,1
1,6,15,20,15,6,1
1,7,21,35,35,21,7,1
Step-by-step explanation:
(a+b)^n
=a^n+na^(n-1)b+(n(n-1)/2)a^(n-2)b²+...+nab^(n-1)+b^n
=C(n,0)a^n+C(n,1)a^(n-1)b+C(n-2)a^(n-2)b^sup2;.....C(n,n)b^n
For n=1
(a+b)1=a+b
(a+b)2=a²+2ab+b^sup2;
(a+b)3=a³+3a^sup2;b+3ab²+b³
...
The coefficients are therefore
1,1
1,2,1
1,3,3,1
1,4,6,4,1
1,5,10,10,5,1
1,6,15,20,15,6,1
1,7,21,35,35,21,7,1
...
This is the Pascal's triangle
The sum of each pair of adjacent numbers gives rise to a number on the next line.
This also means that
(a+b)7
=a^7+7a^6b+21a^5b²+35a^4b³+....+7ab^6+b^7 ............(A)
which is almost the solution to the problem.
To solve the problem using the Pascal's triangle, substitute a=x, and b=2y in equation (A) to give the expression of the final answer.
Answer: The fire is 3.5 miles from tower B
Step-by-step explanation: Please refer to the attached diagram. The triangle in the attached diagram illustrates the clues given in the question. Both rangers are standing at points A and B respectively with a distance of 5 miles between them, which is line AB. Also, one ranger spots a fire from a tower at an angle of 42 degrees, which is point A. Another ranger spots the same fire from another tower, but from an angle of 64 degrees, which is point B. The fire is at point C on the triangle. Now we have a triangle with only one side known (5 miles) and three angles known (the third angle is computed as 180 - {64+42} which equals 74) which are 64 degrees, 42 degrees and 74 degrees.
The distance from the fire to tower B is calculated using the law of sines. (Note that this is not a right angled triangle, hence we cannot use trigonometric ratios). The law of sines is expressed as follows;
a/SinA = b/SinB or
a/SinA = c/SinC
Depending on the sides and angles we are given and the ones we are to calculate.
The distance from the fire to tower B is line BC, labeled as a in our diagram. Using the law of sines
a/SinA = c/SinC
(Note also that a is directly facing angle A, c is directly facing angle C, and so on)
a/SinA = c/SinC
a/Sin 42 = 5/Sin 74
By cross multiplication we now have
a (Sin 74) = 5 (Sin 42)
Divide both sides of the equation by Sin 74 and we now arrive at
a = 5 (Sin 42)/Sin 74
a = 5 (0.6691)/0.9613
a = 3.3455/0.9613
a = 3.4802
{rounded to the nearest tenth of a mile, a equals 3.5}
Therefore the distance from tower B to the fire is approximately 3.5 miles