name: \nteacher: ms. gao\na farmer has some c...
name: \nteacher: ms. gao\na farmer has some cows and horses. all the animals are either brown or black. the table shows how many of each animal is on the farm.\nfarm animals\n| |brown|black|\n|--|--|--|\n|cows|3|5|\n|horses|4|8|\none animal is selected randomly. what is the probability of choosing an animal that is black or of choosing a cow?\na. 0.80\nb. 0.65\nc. 0.40\nd. 0.20\nthe speed of a car is modeled by the graph. at what interval is the function increasing?\nspeed of a car\n(0, 20) and (40, 50)\n(0, 20) and (70, 80)\n(20, 40) and (50, 70)\n(20, 40) and (80, 100)\nwhat is tanθ if θ lies in quadrant 4, and sinθ = -\\frac{1}{2}?\n-\\frac{\\sqrt{3}}{3}\n\\frac{\\sqrt{3}}{3}\n-\\frac{\\sqrt{3}}{5}\n\\frac{\\sqrt{3}}{5}
Answer
### 1. First problem (probability of choosing a cow given black - colored animal)
# Explanation:
## Step1: Calculate total black animals
The number of black cows is 5 and black horses is 8. So total black animals are $5 + 8=13$.
## Step2: Calculate probability
The number of black cows is 5. The probability $P$ of choosing a cow given the animal is black is $\frac{\text{Number of black cows}}{\text{Total number of black animals}}=\frac{5}{13}\approx 0.385\approx0.40$.
# Answer:
C. 0.40
### 2. Second problem (interval of increasing function for car - speed graph)
# Explanation:
## Step1: Recall definition of increasing function
A function $y = f(x)$ is increasing when as $x$ increases, $y$ also increases.
## Step2: Analyze the graph
Looking at the speed - distance graph of the car, the speed (y - value) is increasing in the intervals $(0,20)$ and $(70,80)$.
# Answer:
$(0,20)$ and $(70,80)$
### 3. Third problem (finding $\tan\theta$ given $\sin\theta$ and quadrant)
# Explanation:
## Step1: Find $\cos\theta$ using $\sin^{2}\theta+\cos^{2}\theta = 1$
Given $\sin\theta=-\frac{1}{2}$, then $\cos^{2}\theta=1 - \sin^{2}\theta=1-\left(-\frac{1}{2}\right)^{2}=1-\frac{1}{4}=\frac{3}{4}$. Since $\theta$ is in Quadrant 4, $\cos\theta=\frac{\sqrt{3}}{2}$ (cosine is positive in Quadrant 4).
## Step2: Calculate $\tan\theta$
We know that $\tan\theta=\frac{\sin\theta}{\cos\theta}$. Substituting $\sin\theta =-\frac{1}{2}$ and $\cos\theta=\frac{\sqrt{3}}{2}$, we get $\tan\theta=\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}$.
# Answer:
$-\frac{\sqrt{3}}{3}$