Make the variable given after each equation the subject of the formula: a) \( \quad V=\pi r^{2} h \) for \( h \) b) \( \quad V=\pi r^{2} h \) for \( r \) c) \( \quad A=P(1+i n) \) for \( i \) d) \( \quad A=P(1+i)^{n} \) for \( i \) e) \( 3=\frac{-b+\sqrt{b^{2}-4 a c}}{2 a} \) for c f) \( s=u t+\frac{1}{2} a t^{2} \) for a g) \( s=u t+\frac{1}{2} a t^{2} \) for u h) \( \quad S A=2(l b+l h+b h) \) for h i) \( \quad 0=a x^{2}+b x+c \quad \) for \( a \) j) \( \quad T_{n}=a(r)^{n-1} \) for \( r \)

Below are one acceptable set of solutions. Note that there may be several equivalent ways to rearrange each equation. ───────────────────────────── a) Solve V = πr²h for h Step 1. Divide by πr²:   h = V⁄(πr²) ───────────────────────────── b) Solve V = πr²h for r Step 1. Divide by πh:   r² = V⁄(πh) Step 2. Take the square root (assuming r ≥ 0):   r = √[V⁄(πh)] ───────────────────────────── c) Solve A = P(1 + in) for i Step 1. Divide by P: 1 + in = A⁄P Step 2. Subtract 1: in = (A⁄P) – 1 Step 3. Divide by n: i = [(A⁄P) – 1]⁄n (Alternatively, one can write i = (A – P)/(nP).) ───────────────────────────── d) Solve A = P(1+i)ⁿ for i Step 1. Divide by P: (1+i)ⁿ = A⁄P Step 2. Take the nth root: 1+i = (A⁄P)^(1/n) Step 3. Subtract 1: i = (A⁄P)^(1/n) – 1 ───────────────────────────── e) Solve 3 = (–b + √(b² – 4ac))/(2a) for c Step 1. Multiply both sides by 2a (assuming a ≠ 0):   6a = –b + √(b² – 4ac) Step 2. Add b:   6a + b = √(b² – 4ac) Step 3. Square both sides:   (6a + b)² = b² – 4ac Step 4. Expand and simplify. Expand the left side:   36a² + 12ab + b² = b² – 4ac Step 5. Cancel b² from both sides:   36a² + 12ab = –4ac Step 6. Solve for c by dividing by –4a (assuming a ≠ 0):   c = –(36a² + 12ab)/(4a) = –9a – 3b ───────────────────────────── f) Solve s = ut + (½)at² for a Step 1. Subtract ut:   s – ut = (½)at² Step 2. Multiply by 2:   2(s – ut) = at² Step 3. Divide by t² (t ≠ 0):   a = 2(s – ut)/t² ───────────────────────────── g) Solve s = ut + (½)at² for u Step 1. Subtract (½)at²:   s – (½)at² = ut Step 2. Divide by t (t ≠ 0):   u = [s – (½)at²]/t ───────────────────────────── h) Solve SA = 2(lb + lh + bh) for h Step 1. Divide both sides by 2:   SA⁄2 = lb + lh + bh Step 2. Factor h from the terms that contain it:   SA⁄2 = lb + h(l + b) Step 3. Subtract lb:   SA⁄2 – lb = h(l + b) Step 4. Divide by (l + b) (assuming l+b ≠ 0):   h = [SA⁄2 – lb]⁄(l + b) ───────────────────────────── i) Solve 0 = ax² + bx + c for a Step 1. Write the equation as:   ax² + bx + c = 0 Step 2. Isolate ax²:   ax² = –bx – c Step 3. Divide by x² (assuming x ≠ 0):   a = –(bx + c)/x² ───────────────────────────── j) Solve Tₙ = a·r^(n–1) for r Step 1. Divide by a (a ≠ 0):   r^(n–1) = Tₙ⁄a Step 2. Take the (n–1)th root (provided n–1 ≠ 0):   r = (Tₙ⁄a)^(1/(n–1)) ───────────────────────────── These are the rearranged formulas with the indicated variable as the subject. Any algebraically equivalent form is correct.