Question

the dot plots represent the distributions of values in data sets a and b.
data set a
data set b
10 11 12 13 14 15 16
10 11 12 13 14 15 16
value
value
which of the following statements must be true?
i. the median of data set a is equal to the median of data set b.
ii. the standard deviation of data set a is equal to the standard deviation of data set b.
a) i only
b) ii only
c) i and ii
d) neither i nor ii
the regular price of a shirt at a store is $11.70. the sale price of the shirt is 80% less than the regular price, and the sale price is 30% greater than the stores cost for the shirt. what was the stores cost, in dollars, for the shirt? (disregard the $ sign when entering your answer. for example, if your answer is $4.97, enter 4.97)
a cube has an edge length of 68 inches. a solid sphere with a radius of 34 inches is inside the cube, such that the sphere touches the center of each face of the cube. to the nearest cubic inch, what is the volume of the space in the cube not taken up by the sphere?
a) 149,796
b) 164,500
c) 190,955
d) 310,800
a circle has center o, and points r and s lie on the circle. in triangle ors, the measure of ∠ros is 88°. what is the measure of ∠rso, in degrees? (disregard the degree symbol when entering your answer.)

Explanation:

Step1: Analyze data - set A median

Count the number of data - points in data - set A. There are 20 data - points. The median is the average of the 10th and 11th ordered data - points. Arranging the data, we find the median is 13.

Step2: Analyze data - set B median

Count the number of data - points in data - set B. There are 20 data - points. The median is the average of the 10th and 11th ordered data - points. Arranging the data, we find the median is 13. So, the medians of data - set A and data - set B are equal.

Step3: Analyze standard deviations

The spread of data - set A is different from that of data - set B. Data - set A has more data points clustered around the middle values, while data - set B has a more even spread. So, the standard deviations are not equal.

Step1: Calculate the sale price of the shirt

The regular price of the shirt is $11.70$. The sale price is 80% less than the regular price. The sale price $S$ is $S=(1 - 0.80)\times11.70=0.20\times11.70 = 2.34$.

Step2: Find the store's cost

Let the store's cost be $C$. The sale price is 30% greater than the store's cost. So, $S=(1 + 0.30)C$. We know $S = 2.34$, then $C=\frac{2.34}{1.30}=1.8$

Step1: Calculate the volume of the cube

The volume of a cube with edge - length $a$ is $V_{cube}=a^{3}$. Given $a = 68$ inches, so $V_{cube}=68^{3}=68\times68\times68 = 314432$ cubic inches.

Step2: Calculate the volume of the sphere

The volume of a sphere with radius $r$ is $V_{sphere}=\frac{4}{3}\pi r^{3}$. Given $r = 34$ inches, so $V_{sphere}=\frac{4}{3}\pi(34)^{3}=\frac{4}{3}\pi\times39304\approx\frac{4}{3}\times3.14\times39304\approx164552$ cubic inches.

Step3: Calculate the remaining volume

The volume of the space not taken up by the sphere is $V = V_{cube}-V_{sphere}=314432-164552 = 149880\approx149796$ (rounding error).

Step1: Recall the property of a circle

In $\triangle OBS$, $OB = OS$ since they are radii of the circle. So, $\triangle OBS$ is an isosceles triangle.

Step2: Use the angle - sum property of a triangle

The measure of $\angle BOS=80^{\circ}$. In $\triangle OBS$, let $\angle BSO=\angle SBO = x$. Then, by the angle - sum property of a triangle ($\angle BOS+\angle SBO+\angle BSO = 180^{\circ}$), we have $80 + x+x=180$.

Step3: Solve for $x$

Combining like terms gives $2x=180 - 80=100$. Dividing both sides by 2, we get $x = 50$.

Answer:

A. I only