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olasank [31]
2 years ago
11

Three cell phones towers can be modeled by the points X(6,0), Y(8,4), and Z(3,9). Determine the location of another cell phone t

ower equidistant from the other three, and write an equation for the circle.
Mathematics
1 answer:
Marat540 [252]2 years ago
3 0

Answer:

The location of the new cell phone tower is (h,k) = (3,4), and the equation of the circle is x^{2}+y^{2} -6\cdot x - 8\cdot y = 0.

Step-by-step explanation:

The location of the cell phone tower coincides with the location of a circunference passing through the three cell phone towers. By Analytical Geometry, the equation of the circle is represented by the following general formula:

x^{2} + y^{2}+A\cdot x + B\cdot y +C = 0 (1)

Where:

x - Independent variable.

y - Dependent variable.

A, B, C - Circunference constants.

Given the number of variable, we need the location of three distinct points:

(x_{1},y_{1}) = (6,0)

36 +6\cdot A + C = 0

(x_{2},y_{2}) = (8,4)

80 + 8\cdot A + 4\cdot B + C = 0

(x_{3},y_{3}) = (3,9)

90 + 3\cdot A + 9\cdot B + C = 0

Then, we have the following system of linear equations:

6\cdot A + C = -36 (2)

8\cdot A +4\cdot B + C = -80 (3)

3\cdot A + 9\cdot B + C = -90 (4)

The solution of this system is:

A = -6, B = -8, C = 0

By comparing the general form with the standard form of the equation of the circunference is:

A = -2\cdot h (5)

B = -2\cdot k (6)

C = h^{2}+k^{2}-r^{2} (7)

Where:

h, k - Coordinates of the center of the circle.

r - Radius of the circle.

If we know that A = -6, B = -8 and C = 0, then coordinates of the center of the circle and its radius are, respectively:

h = -\frac{A}{2}

k = -\frac{B}{2}

r = \sqrt{h^{2}+k^{2}-C}

h = 3, k = 4, r = 5

The location of the new cell phone tower is (h,k) = (3,4), and the equation of the circle is x^{2}+y^{2} -6\cdot x - 8\cdot y = 0.

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Answer:

a) 3, 6, 9, 12, 15,...,3\cdot n, b) 4, 7, 10, 13, 16,...,3\cdot n +1, c) Both sequences are arithmetic.

Step-by-step explanation:

a) The sequence of natural numbers which are multiplied by 3 are represented by the function f(n) = 3\cdot n, n\in \mathbb{N}. Let see the first five elements of the sequence: 3, 6, 9, 12, 15,...

b) The sequence of natural numbers which are multiplied by 3 and added to 1 is represented by the function f(n) = 3\cdot n + 1, n\in \mathbb{N}. Let see the first five elements of the sequence: 4, 7, 10, 13, 16,...

c) Both sequences since differences between consecutive elements is constant. Let prove this statement:

(i) f(n) = 3\cdot n

\Delta f = f(n+1) -f(n)

\Delta f = 3\cdot (n+1) -3\cdot n

\Delta f = 3

(ii) f(n) = 3\cdot n +1

\Delta f = f(n+1)-f(n)

\Delta f = [3\cdot (n+1)+1]-(3\cdot n+1)

\Delta f = 3

Both sequences are arithmetic.

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Answer:

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Step-by-step explanation:

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Answer:

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Step-by-step explanation:

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