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1. the behavior of a gas can be approximated by the ideal gas law, $pv = nrt$. in this formula, $p$, $v$, and $t$ are the gas’s pressure, volume, and temperature; $n$ is the amount of the gas in moles; and $r$ is a constant. solve the formula for $t$.
2. the formula for the volume of a right circular cone is shown. rewrite the formula so that it expresses the height $h$ in terms of the volume $v$ and base radius $r$.
image of a cone with height $h$, radius $r$, and formula $v = \frac{1}{3}\pi r^2 h$
3. solve the literal equation $a(bx + c) = d$ for $x$. state any necessary restrictions on the letters representing constants in the equation. then use the solution of the literal equation to solve $-3(2x + 1) = 4$.
4. alma is choosing a new smartphone and cellular service plan. let $p$ be the phone’s price in dollars, $m$ be the cost of the selected plan in dollars per month, and $c$ be alma’s total cost in dollars after time $t$, measured in months of service. write an equation that expresses $c$ in terms of $p$, $m$, and $t$. then solve your equation for $t$.

Question

Accepted Answer

### Problem 1
# Explanation:
## Step1: Start with the Ideal Gas Law formula $ PV = nRT $. We need to isolate $ T $.
To solve for $ T $, we divide both sides of the equation by $ nR $ (assuming $ n
eq0 $ and $ R
eq0 $, since $ n $ is the amount of gas in moles and $ R $ is a constant, these conditions are usually satisfied in the context of ideal gas law).
$$
\frac{PV}{nR}=\frac{nRT}{nR}
$$
## Step2: Simplify the right - hand side.
The $ n $ and $ R $ in the numerator and denominator on the right - hand side cancel out, leaving us with:
$$
T = \frac{PV}{nR}
$$
# Answer:
$ T=\frac{PV}{nR} $

### Problem 2
# Explanation:
## Step1: Start with the volume formula of a right - circular cone $ V=\frac{1}{3}\pi r^{2}h $. We want to solve for $ h $.
First, multiply both sides of the equation by 3 to get rid of the fraction on the right - hand side.
$$
3V=\pi r^{2}h
$$
## Step2: Isolate $ h $.
Divide both sides of the equation by $ \pi r^{2} $ (assuming $ r
eq0 $, because if $ r = 0 $, the cone degenerates to a line and the volume formula is not meaningful in the context of a non - degenerate cone).
$$
h=\frac{3V}{\pi r^{2}}
$$
# Answer:
$ h = \frac{3V}{\pi r^{2}} $

### Problem 3
# Explanation:
## Step1: Start with the equation $ a(bx + c)=d $. First, divide both sides by $ a $ (assuming $ a
eq0 $, because if $ a = 0 $, the left - hand side becomes 0, and if $ d
eq0 $, the equation has no solution; if $ d = 0 $, the equation becomes $ 0=0 $ for all $ x $ when $ a = 0 $).
$$
bx + c=\frac{d}{a}
$$
## Step2: Subtract $ c $ from both sides.
$$
bx=\frac{d}{a}-c=\frac{d - ac}{a}
$$
## Step3: Divide both sides by $ b $ (assuming $ b
eq0 $, because if $ b = 0 $, the left - hand side becomes 0, and if $ \frac{d - ac}{a}
eq0 $, the equation has no solution; if $ \frac{d - ac}{a}=0 $, the equation becomes $ 0 = 0 $ for all $ x $ when $ b = 0 $).
$$
x=\frac{d - ac}{ab}
$$

Now, use this solution to solve $ - 3(2x+1)=4 $. Here, $ a=-3 $, $ b = 2 $, $ c = 1 $, $ d = 4 $.
Substitute these values into the formula $ x=\frac{d - ac}{ab} $:
$$
x=\frac{4-(-3)\times1}{(-3)\times2}=\frac{4 + 3}{-6}=\frac{7}{-6}=-\frac{7}{6}
$$
# Answer:
For the literal equation $ a(bx + c)=d $, $ x=\frac{d - ac}{ab} $ with restrictions $ a
eq0 $ and $ b
eq0 $ (when $ a = 0 $ or $ b = 0 $, the equation has special cases as discussed above). For the equation $ -3(2x + 1)=4 $, $ x=-\frac{7}{6} $

### Problem 4
# Explanation:
## Step1: Write the equation for total cost $ C $.
The total cost $ C $ is the sum of the price of the phone $ p $ and the total cost of the service plan. The service plan costs $ m $ dollars per month for $ t $ months, so the cost of the service plan is $ mt $ dollars. Thus, the equation is:
$$
C=p + mt
$$
## Step2: Solve for $ t $.
First, subtract $ p $ from both sides of the equation:
$$
C - p=mt
$$
Then, divide both sides by $ m $ (assuming $ m
eq0 $, because if $ m = 0 $, the cost of the service plan is 0 and the equation for $ t $ is not well - defined in the context of a monthly service plan with non - zero time).
$$
t=\frac{C - p}{m}
$$
# Answer:
The equation for $ C $ is $ C=p + mt $, and the solution for $ t $ is $ t=\frac{C - p}{m} $ (with the restriction $ m
eq0 $)

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QUESTION IMAGE

Question

Explanation:

Problem 1

Step1: Start with the Ideal Gas Law formula \( PV = nRT \). We need to isolate \( T \).

Step2: Simplify the right - hand side.

Step1: Start with the volume formula of a right - circular cone \( V=\frac{1}{3}\pi r^{2}h \). We want to solve for \( h \).

Step2: Isolate \( h \).

Step1: Start with the equation \( a(bx + c)=d \). First, divide both sides by \( a \) (assuming \( a

Step2: Subtract \( c \) from both sides.

Step3: Divide both sides by \( b \) (assuming \( b

Answer:

Problem 2