Question

1. write the equation of a circle with a center at (-2,3) and a point on the circle at (-2,7).

a. $(x+2)^2+(y-3)^2=16$
b. $(x+2)^2+(y-3)^2=4$
c. $(x-2)^2+(y+3)^2=4$
d. $(x-2)^2+(y+3)^2=16$

1. the measure of angle $\angle aob = \frac{2\pi}{3}$ radians and the radius is 18cm. what is the length of arc ab?

image of a circle with points a, o, b
a. 12
b. $12\pi$
c. 27
d. $27\pi$

1. what is the length of a radius of the circle represented by the equation $x^2 + y^2 - 4x - 4y + 4 = 0$?

a. 2 units
b. 4 units
c. 8 units
d. 16 units

1. mr. miller’s field of vision is 140 degrees in the diagram below.

image of a diagram showing mr. millers position, horizon, and a 140-degree angle with radius 6 miles
from his beach house he can see ships on the horizon up to 6 miles away. to the nearest tenth of a mile, how many miles of the horizon can mr. miller see along the arc of his field of vision?
a. 2.3 miles
b. 4.7 miles
c. 14.7 miles
d. 534.8 miles

1. what is the area of a sector with an arc length of 40 cm and a radius of 12 cm?

a. $24\pi$ cm²
b. 76.4 cm²
c. 191 cm²
d. 240 cm²

Explanation:

Question 38

Step1: Recall the circle equation formula

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h,k)\) is the center and \(r\) is the radius.

Step2: Find the radius

The center is \((-2,3)\) and a point on the circle is \((-2,7)\). The radius is the distance between these two points. Since the \(x\)-coordinates are the same, the distance (radius) is \(|7 - 3| = 4\), so \(r^2 = 16\).

Step3: Substitute center into the formula

The center \((h,k)=(-2,3)\), so the equation is \((x - (-2))^2 + (y - 3)^2 = 16\), which simplifies to \((x + 2)^2 + (y - 3)^2 = 16\).

Step1: Recall the arc length formula

The formula for arc length \(s\) is \(s = r\theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians.

Step2: Substitute the values

Given \(r = 18\) cm and \(\theta=\frac{2\pi}{3}\) radians, we calculate \(s = 18\times\frac{2\pi}{3}\).

Step3: Simplify the expression

\(18\times\frac{2\pi}{3}= 12\pi\) cm.

Step1: Rewrite the circle equation in standard form

Start with \(x^{2}+y^{2}-4x - 4y + 4 = 0\). Group \(x\) and \(y\) terms: \((x^{2}-4x)+(y^{2}-4y)= - 4\).

Step2: Complete the square

For \(x\): \((x^{2}-4x + 4)\) and for \(y\): \((y^{2}-4y + 4)\). Add 4 and 4 to both sides: \((x^{2}-4x + 4)+(y^{2}-4y + 4)= - 4 + 4+4\).

Step3: Write in standard form

This becomes \((x - 2)^2+(y - 2)^2 = 4\). The radius \(r\) is \(\sqrt{4}=2\) units.

Answer:

A. \((x+2)^{2}+(y-3)^{2}=16\)

Question 39