Question

1. devon plans to ride a 15 mi bicycle trail. if his average speed is 20 mi/h, write an equation that can be used to find the time, t, in hours, for the ride.

1. in physics, the ideal gas law describes the relationship among the pressure, volume, and temperature of a gas sample. the law is represented with the formula ( pv = nrt ), where ( p ) is the pressure, ( v ) is the volume, ( t ) is the temperature, ( n ) is the amount of gas, and ( r ) is a physical constant. which of the equations below are equivalent to the formula ( pv = nrt )? select all that apply.

a. ( p = vnrt )
b. ( v = \frac{nrt}{p} )
c. ( n = \frac{pv}{rt} ) (or similar, per ocr)
d. ( r = \frac{pv}{nt} )
e. ( t = \frac{pv}{nr} )

Explanation:

Problem 2:

Step1: Recall the formula for speed

The formula relating speed (\(s\)), distance (\(d\)), and time (\(t\)) is \(s=\frac{d}{t}\). We know the distance \(d = 15\) miles and the speed \(s=20\) miles per hour. We need to solve for \(t\).

Step2: Rearrange the formula to solve for \(t\)

Starting with \(s=\frac{d}{t}\), multiply both sides by \(t\) to get \(s\times t=d\). Then divide both sides by \(s\) to isolate \(t\), so \(t = \frac{d}{s}\).

Step3: Substitute the given values

Substitute \(d = 15\) and \(s = 20\) into the formula \(t=\frac{d}{s}\). So \(t=\frac{15}{20}\). We can simplify this fraction by dividing both the numerator and the denominator by 5, which gives \(t=\frac{3}{4}\) hours or \(t = 0.75\) hours. But the problem just asks for the equation to find \(t\), so from \(s=\frac{d}{t}\), we can write \(t=\frac{d}{s}\), and substituting the values, the equation is \(t=\frac{15}{20}\) (or simplified forms). But if we want the general equation first, it's derived from \(s=\frac{d}{t}\) leading to \(t=\frac{d}{s}\), and then plugging in \(d = 15\) and \(s=20\) gives \(t=\frac{15}{20}\).

(for Problem 3):
To determine equivalent equations to \(PV = nRT\), we rearrange the formula by isolating each variable:

• For \(V\): Divide both sides by \(P\) → \(V=\frac{nRT}{P}\) (matches option b).
• For \(R\): Divide both sides by \(nT\) → \(R=\frac{PV}{nT}\) (matches option d).
• For \(n\): Divide both sides by \(RT\) → \(n=\frac{PV}{RT}\) (if option c has a typo, e.g., \(W\) is \(R\), it’s correct).
• For \(T\): Divide both sides by \(nR\) → \(T=\frac{PV}{nR}\) (if option e has a typo, e.g., \(W\) is \(R\), it’s correct).
• Option a is incorrect as dividing by \(V\) gives \(P=\frac{nRT}{V}\), not \(P = VnRT\).

Answer:

The equation to find the time \(t\) (in hours) is \(t=\frac{15}{20}\) (or \(t=\frac{d}{s}\) with \(d = 15\) and \(s = 20\))

Problem 3:

We start with the ideal gas law \(PV=nRT\). We need to check which of the given equations are equivalent by rearranging the original formula.

• Option a: \(P = VnRT\)

Starting from \(PV=nRT\), if we divide both sides by \(V\), we get \(P=\frac{nRT}{V}\), not \(P = VnRT\). So this is incorrect.

• Option b: \(V=\frac{nRT}{P}\)

Starting from \(PV=nRT\), divide both sides by \(P\): \(V=\frac{nRT}{P}\). This matches, so this is correct.

• Option c: \(n=\frac{PV}{RT}\) (Assuming the typo, maybe \(n=\frac{PV}{RT}\) instead of \(n=\frac{PV}{WT}\))

Starting from \(PV=nRT\), divide both sides by \(RT\): \(n=\frac{PV}{RT}\). If we assume a typo and \(W\) is \(R\), then this is correct. But as written with \(W\), it's incorrect. However, if we consider the ideal gas law, the correct rearrangement for \(n\) is \(n=\frac{PV}{RT}\). Maybe a typo in the problem, but if we proceed with the ideal gas law, the correct form is \(n=\frac{PV}{RT}\).

• Option d: \(R=\frac{PV}{nT}\)

Starting from \(PV=nRT\), divide both sides by \(nT\): \(R=\frac{PV}{nT}\). This matches, so this is correct.

• Option e: \(T=\frac{PV}{nR}\) (Assuming the typo, maybe \(T=\frac{PV}{nR}\) instead of \(T=\frac{PV}{nW}\))

Starting from \(PV=nRT\), divide both sides by \(nR\): \(T=\frac{PV}{nR}\). If we assume a typo and \(W\) is \(R\), then this is correct.