Question 1
\((x-6)^{2}+(y+5)^{2}=16\)
In the \(xy-\)plane, the graph of the equation above is a circle. Point \(P\) is on the circle and has coordinates \((10,-5)\). If \(\overline{P Q}\) is a diameter of the circle, what are the coordinates of point \(Q\)?
A) \((2,-5)\)
B) \((6,-1)\)
C) \((6,-5)\)
D) \((6,-9)\)
Choice A is correct. The standard form for the equation of a circle is \((x-h^{2})+(y-k)^{2}\), where \((h,k\)) are the coordinates of the centre and \(r\) is the length of the radius. According to the given equation, the centre of the circle is \((6,-5)\). Let \((x_{1},y_{1})\) represents the coordinates of point \(Q\). Since point \(P(10,-5)\) and point \(Q(x_{1},y_{1})\) are the endpoints of the diameter of the circle, the centre \((6,-5)\) lies on the diameter halfway between \(P\) and \(Q\). Therefore, the following relationships hold:
\(\frac{x_{1}+10}{2}=6\) and \(\frac{y_{1}+(-5)}{2}=-5\)
Solving the equations for \(x_{1}\) and \(y_{1}\), respectively, yields \(x_{1}=2\) and \(y_{1}=-5\). Therefore, the coordinates of point \(Q\) are \((2,-5)\).
Alternative approach: Since point \(P(10,-5)\) on the circle and the centre of the circle \((6,-5)\) have the same \(y\)-coordinates, it follows that the diameter \(\overline{P Q}\) must have the same \(y\)-coordinates as \(P\) and be \(4\) units away from the centre. Hence, the coordinates of point \(Q\) must be \((2,-5)\).
Choice B and D are incorrect because the points given in these choices lie on a diameter that is perpendicular to the diameter \(\overline{P Q}\). If either of these points were point \(Q\), then \(\overline{P Q}\) would not be the diameter of the circle.
Choice C is incorrect because\((6,-5)\) is the centre of the circle and does not lie on the circle.
Question 2
A group of \(202\) people went on an overnight camping trip, taking \(60\) tents with them. Some tents held \(2\) people each, and the rest held \(4\) people each. Assuming all the tents were filled to capacity, and every person got to sleep in a tent, exactly how many of the tents were \(2\)-person tents?A) \(30\)
B) \(20\)
C) \(19\)
D) \(18\)
Choice C is correct. Let \(x\) represent the number of \(2\)-person tents and let \(y\) represent the number \(4\)-person tents. It is given that the total number of tents was \(60\), and the total number of people in the group was \(202\). This situation can be expressed as a system of two equations, \(x+y=60\) and \(2x+4y=202\).
The first equation can be rewritten as \(y=-x+60\). Subtracting \(-x++60\) for \(y\) in the equation \(2x+4y=202\) yields \(2x+4(-x+60)=202\).
Distributing and combining like terms gives \(-2x+240=202\) and then dividing both sides by \(-2\) gives \(x=19\). Therefore, the number of \(2\)-person tents is \(19\).
Alternate approach: If each of the 60 tents held \(4\) people, the total number of people that could be accommodated in tents would be \(240\). However, the actual number of people who slept in tents was \(202\). The difference of \(38\) accounts for the \(2\)-person tents. Since each of these tents hold \(2\) people fewer than a \(4\)-person tent, \(\frac{38}{2}=19\) gives the number of \(2\)-person tents.
Question 3
The correct answer is 144. In a circle, the ratio of the length of a given arc to the circle’s circumference is equal to the ratio of the measure of the arc, in degrees, to \(360^{\large\circ}\). The ratio between the arc length and the circle’s circumference is given as \(\frac{2}{5}\). It follows that \(\frac{2}{5}=\frac{x}{360}\). Solving this proportion for \(x\) gives \(x=144\).
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