Question

1. if $\frac{y}{x}=3$ and $x

eq0$, which of the following is equal to $y$?
a) $\frac{1}{x}$
b) 1
c) 3
d) $y$ is undefined.

1. a segment of a circle with a circumference of $20pi$ units is bounded by two radii and an arc that measures $4pi$ units. what is the area bounded by these two radii and arc?

a) $20pi$ units²
b) 25 units²
c) $25pi$ units²
d) $80pi$ units²

Explanation:

1. Solve the first - question

Step1: Given equation

We are given $\frac{y}{z}=3$ and $z
eq0$.

Step2: Isolate y

Multiply both sides of the equation $\frac{y}{z}=3$ by $z$. We get $y = 3z$. But if we want to express $y$ in terms of the given options, we can rewrite the equation $\frac{y}{z}=3$ as $y = 3z$. When we consider the relationship between $y$ and $z$ in the form of the options, from $\frac{y}{z}=3$, we know that $y$ is 3 times $z$.

Step1: Find the radius of the circle

The formula for the circumference of a circle is $C = 2\pi r$. Given $C=20\pi$. Then $2\pi r=20\pi$. Divide both sides by $2\pi$, we get $r = 10$ units.

Step2: Find the area of the sector

The formula for the length of an arc $l$ is $l=\theta r$ (where $\theta$ is the central angle in radians). Given $l = 4\pi$ and $r = 10$, then $\theta=\frac{l}{r}=\frac{4\pi}{10}=\frac{2\pi}{5}$ radians. The formula for the area of a sector of a circle is $A=\frac{1}{2}r^{2}\theta$. Substitute $r = 10$ and $\theta=\frac{2\pi}{5}$ into the formula: $A=\frac{1}{2}\times10^{2}\times\frac{2\pi}{5}=\frac{1}{2}\times100\times\frac{2\pi}{5}=20\pi$ square units.

Answer:

C. 3z (assuming the options were mis - typed and the intention was to have an expression related to the given equation. If we assume the options were correct as written, there is an error in the problem setup. But if we just consider the value of the coefficient of $z$ in the relationship $y = 3z$, the closest interpretation is that the value related to the ratio is 3, so C)

2. Solve the second - question