Expressions, Equations, and Relationships

The ladder is the hypotenuse, and the wall height and ground distance are the legs. Use $a^2 + b^2 = c^2$ with $c = 13$ and a leg $a = 5$. Then $b = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$ feet. This models a common construction setup for checking heights safely.

Equal costs means set Plan A's expression equal to Plan B's: $15 + 4.5x = 9 + 6x$. Choice B drops the variable on 4.5, C omits the monthly fee for Plan A, and D is an inequality instead of an equation.

Set the equations equal: $2x+3=-x+9 \Rightarrow 3x=6 \Rightarrow x=2$. Then $y=2(2)+3=7$. Check: In $y=2x+3$, $y=7$; in $y=-x+9$, $y=-2+9=7$. So $(2,7)$ satisfies both. The intersection is the $(x,y)$ that makes both equations true.

For a cylinder, $V = Bh$ where $B = \pi r^2$. The diameter is 6 in, so $r = 3$ in. Compute $B = \pi r^2 = 3.14 \times 3^2 = 3.14 \times 9 = 28.26$. Then $V = Bh = 28.26 \times 8 = 226.08 \approx 226.1$. The volume is 226.1 in^3.

Subtract $3x$ from both sides: $2x - 8 = 12$. Add 8 to both sides: $2x = 20$. Divide by 2: $x = 10$. Check: Left side $= 5(10) - 8 = 50 - 8 = 42$; right side $= 3(10) + 12 = 30 + 12 = 42$. Both sides match, so $x=10$.

For a cylinder, $V = Bh$ where $B = \pi r^2$. The diameter is 6 cm, so $r = 3$ cm. Compute $B = \pi r^2 = 3.14 \times 3^2 = 3.14 \times 9 = 28.26$. Then $V = Bh = 28.26 \times 15 = 423.9$. The volume is 423.9 cm^3.

The diagonal is the hypotenuse of a right triangle with legs 60 and 80. $d = \sqrt{60^2 + 80^2} = \sqrt{3600 + 6400} = \sqrt{10000} = 100$ meters. This is useful in surveying and planning straight paths across rectangular lots.

Equal total cost means set Plan 1's total equal to Plan 2's: $25 + 12x = 10 + 15x$. A drops the variable on 12, B omits the joining fee for Plan 1, and D uses an inequality instead of an equation.

Let $m$ be miles. Set costs equal: $15 + 0.35m = 5 + 0.45m$. Subtract $0.35m$: $15 = 5 + 0.10m$. Subtract 5: $10 = 0.10m$. Divide by \$0.10$: $m = 100$. Check: Plan A $= 15 + 0.35(100) = 15 + 35 = 50$; Plan B $= 5 + 0.45(100) = 5 + 45 = 50$. Equal at $100$ miles.

Both lines have the same slope $m=3$ but different $y$-intercepts ($-4$ and $1$), so they are parallel and never meet. No $(x,y)$ satisfies both at once. Therefore, there is no solution (no intersection).