Question

27 multiple choice 2.5 points find the value of x. round measures to the nearest tenth (one decimal place). triangle image with x, 34°, 18, right angle options: 14.9, 21.7, 29.8, 30.8 28 multiple choice 2.5 points the venn diagram below gives the number of people surveyed who like country, rap, and rock music. venn diagram with rock, rap, country circles and numbers 18, 11, 24, 8, 7, 1, 17 if one person is chosen at random, what is the probability that the person will like country and rock music? options: 15/86, 53/86, 33/86, 1/86

Explanation:

Question 27

Step1: Identify the trigonometric ratio

The triangle is right - angled, and we know an angle (\(34^{\circ}\)) and the adjacent side to the angle (length = 18), and we need to find the hypotenuse \(x\). The cosine of an angle in a right - triangle is defined as \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). So, \(\cos(34^{\circ})=\frac{18}{x}\).

Step2: Solve for \(x\)

We can re - arrange the formula to solve for \(x\). From \(\cos(34^{\circ})=\frac{18}{x}\), we get \(x = \frac{18}{\cos(34^{\circ})}\). We know that \(\cos(34^{\circ})\approx0.8290\). Then \(x=\frac{18}{0.8290}\approx21.7\).

Step1: Find the number of people who like country and rock music

To find the number of people who like both country and rock music, we need to sum the number of people in the intersection of the country and rock circles. The intersection of country and rock includes the region where only country and rock overlap (\(8\)) and the region where all three (country, rock, rap) overlap (\(7\)). So the number of people who like country and rock music is \(8 + 7=15\).

Step2: Find the total number of people surveyed

To find the total number of people surveyed, we sum up all the regions in the Venn diagram. The regions are: Rock only (\(18\)), Rap only (\(24\)), Country only (\(17\)), Rock and Rap only (\(11\)), Rock and Country only (\(8\)), Rap and Country only (\(1\)), and all three (\(7\)). So the total number of people \(N=18 + 24+17 + 11+8 + 1+7=86\).

Step3: Calculate the probability

The probability that a randomly chosen person likes country and rock music is the number of people who like country and rock music divided by the total number of people surveyed. So the probability \(P=\frac{15}{86}\).

Answer:

21.7

Question 28