[Astronomy] Kepler’s Third Law: The square of the time, $T$, required for a planet to orbit the Sun is directly proportional to the cube of the mean distance, $a$, that the planet is from the Sun.

Part A. Write the equation that describes time $T$ (years) in terms of the mean distance, $a$ (AUs). Use $k$ for your constant.

$T(a) = \answer {\sage {k}}a^{\answer {3/2}}$

Part B. Assume that Mars’ mean distance from the Sun is $\sage {a1}$ AUs and it takes Mars about $\sage {T1}$ months to orbit the Sun. Write the equation that describes time $T$ (years) in terms of the mean distance, $a$ (AUs).

$T(a) = \answer [tolerance=0.05]{\sage {k1}}a^{\answer {3/2}}$

[Chemistry] Ideal Gas Law: The product of pressure, $P$ (atmospheres), and volume, $V$ (liters), of a gas is directly proportional to the product of the amount of substance, $n$ (moles), and temperature, $T$ (Celsius).

Part A. Write the equation that describes the Ideal Gas Law. Use $R$ for your constant.

$\answer {\sage {P*V}} = \answer {\sage {n*R*T}}$

Part B. Assume that the temperature, $\sage {temp2} C$, and volume of the gas, $\sage {volume2}$ liters, remain constant. At $\sage {pressure2}$ atmosphere of pressure, there are $\sage {moles2B}$ moles of the gas. Write an equation that describes the pressure on the gas in terms of the amount of the gas.

$P(n) = \answer [tolerance=0.5]{\sage {coefficientB2}} n^{\answer {1}}$

Part C. Now assume that the pressure, $\sage {pressure2}$ atmospheres, and volume of the gas, $\sage {volume2}$ liters, remain constant. At $\sage {temp2C} C$, there are $\sage {moles2C}$ moles of the gas. Write an equation that describes the temperature of the gas in terms of the amount of the gas.

$T(n) = \answer [tolerance=0.05]{\sage {coefficientC2}} n^{\answer {-1}}$

[Physics] The rate of vibration of a string under constant tension, $r$ cm/s, varies inversely with the length of the string, $l$ cm. When the string is $\sage {l3}$ mm long, the rate of vibration is $\sage {r3}$ cm/s. Write an equation that describes the rate of vibration of a string in terms of the length of the string.

$r(l) = \answer {\sage {k3}} l^{\answer {-1}}$

[Physics] The kinetic energy $K$(J) of a moving object varies jointly with its mass $m$ (kg)and the square of its velocity $v$ (m/s). A $\sage {tons4}$ ton car traveling at $\sage {mph4}$ miles per hour has about $\sage {energy4}$ Joules of kinetic energy. Write an equation that describes the amount of kinetic energy, $K$ in terms of mass $m$ and velocity $v$.

$K = \answer [tolerance=0.05]{1/2} m^{\answer {1}} v^{\answer {2}}$