QUESTION 1
The given circle has equation:
We can rewrite to obtain;
This is equation is now in the form;
Where (h,k)=(0,0) is the center of the circle and is the radius.
QUESTION 2
The center of the circle is at (1,1) and passes through the point (4,5).
The radius of this circle is given by;
units.
We substitute into the formula;
Expand if you wish;
Answer:
(a) The two ordered pairs are (0 , 340) and (4 , 285)
(b) The slope is m = -55/4
The slope means the rate of decreases of the owl population was 55/4
per year (P decreased by 55/4 each year)
(c) The model equation is P = -55/4 t + 340
(d) The owl population in 2022 will be 216
(e) At year 2038 will be no more owl in the park
Step-by-step explanation:
* Lets explain how to solve the problem
- The owl population in 2013 was measured to be 340
- In 2017 the owl population was measured again to be 285
- The owl population is P and the time is t where t measure the numbers
of years since 2013
(a) Let t represented by the x-coordinates of the order pairs and P
represented by the y-coordinates of the order pairs
∵ t is measured since 2013
∴ At 2013 ⇒ t = 0
∵ The population P in 2013 was 340
∴ The first order pair is (0 , 340)
∵ The time from 2013 to 2013 = 2017 - 2013 = 4 years
∴ At 2017 ⇒ t = 4
∵ The population at 2017 is 285
∴ The second order pair is (4 , 285)
* The two ordered pairs are (0 , 340) and (4 , 285)
(b) The slope of any lines whose endpoints are (x1 , y1) and (x2 , y2)
is
∵ (x1 , y1) is (0 , 340) and (x2 , y2) is (4 , 285)
∴ x1 = 0 , x2 = 4 and y1 = 340 , y2 = 285
∴
* The slope is m = -55/4
∵ The slope is negative value
∴ The relation is decreasing
* The slope means the rate of decreases of the owl population was
55/4 per year (<em>P decreased by 55/4 each year</em>)
(c) The linear equation form is y = mx + c, where m is the slope and c is
the value of y when x = 0
∵ The population is P and represented by y
∵ The time is t and represented by t
∴ P = mt + c , c is the initial amount of population
∵ m = -55/4
∵ The initial amount of the population is 340
∴ P = -55/4 t + 340
* The model equation is P = -55/4 t + 340
(d) Lets calculate the time from 2013 to 2022
∵ t = 2022 - 2013 = 9 years
∵ P = -55/4 t + 340
∴ P = -55/4 (9) + 340 = 216.25 ≅ 216
* The owl population in 2022 will be 216
(e) If the model is accurate , then the owl population be be zero after
t years
∵ P = -55/4 t + 340
∵ P = 0
∴ 0 = -55/4 t + 340
- Add 55/4 t to both sides
∴ 55/4 t = 340
- Multiply both sides by 4
∴ 55 t = 1360
- Divide both sides by 55
∴ t = 24.7 ≅ 25 years
- To find the year add 25 years to 2013
∵ 2013 + 25 = 2038
* At year 2038 will be no more owl in the park
The square of a binomial expands as follows:
Expanding all the binomials, we have
Sum like terms:
Simplify terms appearing on both sides:
Move all terms involving k to the left and all numbers to the right:
Divide both sides by -20: