Question

the circle shown has center o, circumference 144π, and diameters (overline{pr}) and (overline{qs}). the length of arc ps is twice the length of arc pq. what is the length of arc qr? a) 24π b) 48π c) 72π d) 96π the frequency table summarizes the 57 data values in a data set. what is the maximum data value in the data set? one leg of a right - triangle has a length of 43.2 millimeters. the hypotenuse of the triangle has a length of 196.8 millimeters. what is the length of the other leg of the triangle, in millimeters? a) 43.2 b) 120 c) 192 d) 201.5

Explanation:

Question 14

Step1: Analyze frequency - table data

The left - hand column represents data values. The largest value in the left - hand column is 14.

Step1: Recall the Pythagorean theorem

The Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse and \(a\) and \(b\) are the legs of a right - triangle. Let one leg \(a = 43.2\) and the hypotenuse \(c=196.8\). We want to find the other leg \(b\).

Step2: Rearrange the Pythagorean theorem

We get \(b=\sqrt{c^{2}-a^{2}}\). Substitute \(a = 43.2\) and \(c = 196.8\) into the formula: \(b=\sqrt{(196.8)^{2}-(43.2)^{2}}\).

Step3: Calculate \(c^{2}-a^{2}\)

\((196.8)^{2}-(43.2)^{2}=(196.8 + 43.2)(196.8-43.2)\) (using the difference - of - squares formula \(x^{2}-y^{2}=(x + y)(x - y)\)). So \((196.8 + 43.2)(196.8-43.2)=240\times153.6\).
\(240\times153.6 = 36864\).

Step4: Calculate the square root

\(b=\sqrt{36864}=192\).

Step1: Recall the property of a circle's circumference

The circumference of a circle \(C = 144\pi\). The sum of the lengths of the arcs of a circle is equal to its circumference. Let the length of arc \(PQ=x\), then the length of arc \(PS = 2x\). Since the circle is divided into four arcs \(PQ\), \(QR\), \(RS\), and \(SP\) and diameters \(PR\) and \(QS\) intersect at the center of the circle, arc \(PQ\) and arc \(RS\) are equal, and arc \(PS\) and arc \(QR\) are equal.

Step2: Set up an equation for the circumference

\(x + 2x+x + 2x=144\pi\). Combining like terms, we get \(6x = 144\pi\).

Step3: Solve for \(x\)

\(x=\frac{144\pi}{6}=24\pi\).

Step4: Find the length of arc \(QR\)

Since the length of arc \(QR\) is \(2x\), substituting \(x = 24\pi\), we get \(2x=48\pi\).

Answer:

14

Question 15