91Ó°ÊÓ

Plot each point and form the triangle \(A B C\). Show that the triangle is a right triangle. Find its area. $$ A=(-2,5) ; \quad B=(1,3) ; \quad C=(-1,0) $$

In Problems 33-38, a point on a line and its slope are given. Find the point- slope form of the equation of the line. $$ P=(1,2) ; m=3 $$

The volume \(V\) of a gas held at a constant temperature in a closed container varies inversely with its pressure \(P .\) If the volume of a gas is 600 cubic centimeters \(\left(\mathrm{cm}^{3}\right)\) when the pressure is 150 millimeters of mercury (mm \(\mathrm{Hg}\) ), find the volume when the pressure is \(200 \mathrm{~mm} \mathrm{Hg}\).

The intensity \(I\) of light (measured in foot-candles) varies inversely with the square of the distance from the bulb. Suppose that the intensity of a 100 -watt light bulb at a distance of 2 meters is 0.075 foot-candle. Determine the intensity of the bulb at a distance of 5 meters.

Find the standard form of the equation of each circle. Center (1,0) and containing the point (-3,2)

The horsepower (hp) that a shaft can safely transmit varies directly with its speed (in revolutions per minute, rpm) and the cube of its diameter. If a shaft of a certain material 2 inches in diameter can transmit 36 hp at \(75 \mathrm{rpm},\) what diameter must the shaft have in order to transmit 45 hp at 125 rpm?

Gas Laws The volume \(V\) of an ideal gas varies directly with the temperature \(T\) and inversely with the pressure \(P\). Write an equation relating \(V, T,\) and \(P\) using \(\underline{k}\) as the constant of proportionality. If a cylinder contains oxygen at a temperature of \(300 \mathrm{~K}\) and a pressure of 15 atmospheres in a volume of 100 liters, what is the constant of proportionality \(k ?\) If a piston is lowered into the cylinder, decreasing the volume occupied by the gas to 80 liters and raising the temperature to \(310 \mathrm{~K},\) what is the gas pressure?

Find the standard form of the equation of each circle. Center (2,-4) and circumference \(16 \pi\)

The electrical resistance of a wire varies directly with the length of the wire and inversely with the square of the diameter of the wire. If a wire 432 feet long and 4 millimeters in diameter has a resistance of 1.24 ohms, find the length of a wire of the same material whose resistance is 1.44 ohms and whose diameter is 3 millimeters.

Kinetic Energy The kinetic energy \(K\) of a moving object varies directly with its mass \(m\) and the square of its velocity \(v\). If an object weighing 25 kilograms and moving with a velocity of 10 meters per second has a kinetic energy of 1250 joules, find its kinetic energy when the velocity is 15 meters per second.