Question

1. g.gsr.8.3: which of the points does not lie on the circle $(x + 2)^2 + (x - 4)^2 = 25$?

a $(-2, -1)$
b $(1, 8)$
c $(3, 4)$
d $(0, 5)$

1. g.gsr.7.2: through how many radians does the minutes hand of a clock turn in 24 minutes?

a $0.2\pi$
b $0.4\pi$
c $0.6\pi$
d $0.8\pi$

1. g.gsr.7.2: sue plays vinyl records on a turntable that rotates through an angle of $\frac{3\pi}{2}$ radians in one second. how many revolutions does the turntable make in one minute?

a $33\frac{1}{3}$ revolutions per minute

Explanation:

Question 9

Step1: Recall circle equation

The standard circle equation is \((x - h)^2 + (y - k)^2 = r^2\), here the circle is \((x + 2)^2 + (y - 4)^2 = 25\) (assuming a typo, should be \((y - 4)^2\) instead of \((x - 4)^2\) as per standard form), so center \((-2, 4)\), radius \(5\).

Step2: Check point A \((-2, -1)\)

Substitute into equation: \((-2 + 2)^2 + (-1 - 4)^2 = 0 + 25 = 25\), lies on circle.

Step3: Check point B \((1, 8)\)

\((1 + 2)^2 + (8 - 4)^2 = 9 + 16 = 25\), lies on circle.

Step4: Check point C \((3, 4)\)

\((3 + 2)^2 + (4 - 4)^2 = 25 + 0 = 25\), lies on circle.

Step5: Check point D \((0, 5)\)

\((0 + 2)^2 + (5 - 4)^2 = 4 + 1 = 5
eq 25\), does not lie on circle.

Step1: Minutes hand rotation

A clock's minutes hand makes a full rotation (\(2\pi\) radians) in 60 minutes.

Step2: Calculate radians per minute

Radians per minute: \(\frac{2\pi}{60} = \frac{\pi}{30}\) radians per minute.

Step3: Radians in 24 minutes

For 24 minutes: \(24\times\frac{\pi}{30} = \frac{4\pi}{5} = 0.8\pi\)? Wait, no, wait: Wait, \(\frac{24}{60} = 0.4\), so \(0.4\times2\pi = 0.8\pi\)? Wait, no, full rotation is \(2\pi\) in 60 min. So fraction of rotation in 24 min: \(\frac{24}{60} = 0.4\). So radians: \(0.4\times2\pi = 0.8\pi\)? Wait, no, wait, no: Wait, the minutes hand moves \(2\pi\) radians in 60 minutes. So in 1 minute, it moves \(\frac{2\pi}{60}=\frac{\pi}{30}\) radians. In 24 minutes: \(24\times\frac{\pi}{30}=\frac{4\pi}{5}=0.8\pi\)? Wait, but let's recalculate: \(\frac{24}{60}=\frac{2}{5}\), so \(\frac{2}{5}\times2\pi=\frac{4\pi}{5}=0.8\pi\). Wait, but the options have 0.8π as D? Wait, no, wait, maybe I messed up. Wait, 60 minutes is \(2\pi\) radians. So 1 minute is \(\frac{2\pi}{60}=\frac{\pi}{30}\) radians. 24 minutes: \(24\times\frac{\pi}{30}=\frac{4\pi}{5}=0.8\pi\). So the answer is D? Wait, no, wait, the options are A:0.2π, B:0.4π, C:0.6π, D:0.8π. Wait, 24/60 = 0.4, so 0.4 of a full rotation. A full rotation is \(2\pi\), so 0.4*2π = 0.8π. So yes, D.

Step1: Determine full rotation time

Minutes hand rotates \(2\pi\) radians in 60 minutes.

Step2: Find fraction of 60 minutes

24 minutes is \(\frac{24}{60} = 0.4\) of 60 minutes.

Step3: Calculate radians

Radians = \(0.4\times2\pi = 0.8\pi\).

Step1: Radians per second

Turntable rotates \(\frac{3\pi}{2}\) radians per second.

Step2: Radians per minute

In 1 minute (60 seconds), radians: \(60\times\frac{3\pi}{2} = 90\pi\) radians.

Step3: Revolutions per minute

One revolution is \(2\pi\) radians. So revolutions: \(\frac{90\pi}{2\pi} = 45\)? Wait, but option A is \(33\frac{1}{3}\). Wait, maybe I made a mistake. Wait, \(\frac{3\pi}{2}\) radians per second. In 60 seconds: \(60\times\frac{3\pi}{2}=90\pi\) radians. Revolutions: \(90\pi\div2\pi = 45\). But option A is \(33\frac{1}{3}\). Wait, maybe the problem was \(\frac{\pi}{3}\) radians? Wait, no, the problem says \(\frac{3\pi}{2}\). Wait, maybe a typo? Wait, if it's \(\frac{\pi}{3}\) radians per second, then 60 seconds: \(60\times\frac{\pi}{3}=20\pi\), revolutions: \(20\pi\div2\pi = 10\). No. Wait, maybe the original problem is \(\frac{5\pi}{3}\)? No. Wait, the option A is \(33\frac{1}{3}\), which is \(\frac{100}{3}\). Let's see: \(\frac{3\pi}{2}\) radians per second. In 60 seconds: \(60\times\frac{3\pi}{2}=90\pi\). Revolutions: \(90\pi\div2\pi = 45\). But 45 is not an option. Wait, maybe the problem was \(\frac{5\pi}{6}\) radians per second? No. Wait, maybe I misread the problem. Wait, the problem says "rotates through an angle of \(\frac{3\pi}{2}\) radians in one second". Wait, \(\frac{3\pi}{2}\) radians per second. So in 60 seconds, \(60\times\frac{3\pi}{2}=90\pi\) radians. One revolution is \(2\pi\) radians, so \(90\pi\div2\pi = 45\) revolutions per minute. But the option A is \(33\frac{1}{3}\). Wait, maybe the problem was \(\frac{5\pi}{3}\) radians per second? No. Wait, maybe a mistake in the problem. But assuming the problem is correct as given, and maybe I made a mistake. Wait, \(33\frac{1}{3}\) is \(\frac{100}{3}\). Let's see: \(\frac{100}{3}\) revolutions per minute. Each revolution is \(2\pi\) radians, so total radians per minute: \(\frac{100}{3}\times2\pi=\frac{200\pi}{3}\). Radians per second: \(\frac{200\pi}{3}\div60=\frac{10\pi}{9}\approx3.49\) radians per second. But \(\frac{3\pi}{2}=4.71\) radians per second. So that's not matching. Wait, maybe the problem was \(\frac{\pi}{3}\) radians per second? Then radians per minute: \(60\times\frac{\pi}{3}=20\pi\), revolutions: \(20\pi\div2\pi = 10\). No. Wait, maybe the original problem has a typo, and the angle is \(\frac{5\pi}{6}\) radians per second? Then radians per minute: \(60\times\frac{5\pi}{6}=50\pi\), revolutions: \(50\pi\div2\pi = 25\). No. Alternatively, maybe the problem is \(\frac{3\pi}{2}\) radians per minute? No, the problem says "in one second". Wait, maybe I misread the exponent. Wait, the problem says \(\frac{3\pi}{2}\) radians in one second. So 60 seconds: \(60\times\frac{3\pi}{2}=90\pi\) radians. Revolutions: \(90\pi\div2\pi = 45\). But 45 is not an option. The option A is \(33\frac{1}{3}\), which is \(\frac{100}{3}\approx33.33\). Let's check: \(\frac{100}{3}\) revolutions per minute. Radians per minute: \(\frac{100}{3}\times2\pi=\frac{200\pi}{3}\). Radians per second: \(\frac{200\pi}{3}\div60=\frac{10\pi}{9}\approx3.49\) radians per second. But \(\frac{3\pi}{2}=4.71\) radians per second. So that's not matching. Maybe the problem is correct, and the option is wrong, or I made a mistake. Wait, maybe the circle equation in question 9 was misread. Let's recheck question 9. The original problem says \((x + 2)^2 + (x - 4)^2 = 25\)? That can't be, it should be \((x + 2)^2 + (y - 4)^2 = 25\). So assuming that, then point D \((0,5)\): \((0 + 2)^2 + (5 - 4)^2 = 4 + 1 = 5
eq 25\), so D is correct. For question 10, the minutes hand: 60 minutes is \(2\p…

Answer:

D. \((0, 5)\)

Question 10