Algebra and Calculus Problem Solutions
Posted on Jan 31, 2026 in Mathematics
Mathematical Problem Set 1
- Profit Function: P(x) = -4x2 + 632x – 4000, where x > 10 and x is the number of units produced. Find the total production for $800 profit.
Work: 0 = -4x2 + 632x – 4800. Plug into the quadratic formula.
ANS: x = 150 - Relative Maximum: F(x) = ln(2x – 8.4) – 0.25(x + 4)2 + 9.83.
ANS: 10.2 at x = 5.5 - Real Zeros: 3 – 2 log5 √(x2 + 2x + 3), to three decimal places.
ANS: x = -12.091, x = 10.091 - Carbon-14 Dating: A 2,500-year-old estimate of Carbon-14 shows a decrease from 20 grams to 12 grams, so 60% remains. P(t) = 20e-0.00012t.
ANS: t = 4256.8 years - Projectile Motion: 72 feet per second, h(t) = -16t2 + 72t.
Work: x = -b/2a ⇒ x = -72/-32.
ANS: Max of 81 ft - Alcohol Consumption: A(x) = -0.018x3 + 1.09x. Are they drunk after 5 hours?
ANS: A(5) = -0.018(5)3 + 1.09(5) = 3.2. Yes, they are drunk. - Rate of Change: Since 2007, y = 0.0238x + 13.4157. What is the rate of change?
ANS: Slope = 0.0238 or 119/5000 - Exponential Growth: 4.8% growth rate since 2008, starting population 350. Find the population in 2012.
ANS: Equation 350e0.048t, P(4) = 424 - Savings Account: $2,500 savings account, interest grows continuously. How long until it reaches $3,300?
Work: 3300 = 2500e0.023t ⇒ 1.32 = e0.023t ⇒ ln(1.32) = 0.023t.
ANS: t = 21.4 years - Logarithmic Equation: log2(x – 3) + log2(x – 5) = 3.
Work: log2(x2 – 5x – 3x + 15) = 3. Multiply both sides by 2 (base conversion) ⇒ x2 – 8x + 15 = 8 ⇒ x2 – 8x + 7 = 0 ⇒ (x – 7)(x – 1).
ANS: x = 7 - Difference Quotient: Find the difference quotient of f(x) = x2 – 6x + 8.
Work: f(x) = [(x + h)2 – 6(x + h) + 8 – (x2 – 6x + 8)] / h. Cancel out terms.
ANS: 2x + h – 6 - Inequality: (x – 6) / (x + 3) ≥ 6.
ANS: [6, ∞)
Mathematical Problem Set 2
- Profit Function: P(x) = -4x2 + 632x – 4000, where x > 10 and x is the number of units produced. Find the total production for $800 profit.
Work: 0 = -4x2 + 632x – 4800. Plug into the quadratic formula.
ANS: x = 150 - Relative Maximum: F(x) = ln(2x – 8.4) – 0.25(x + 4)2 + 9.83.
ANS: 10.2 at x = 5.5 - Real Zeros: 3 – 2 log5 √(x2 + 2x + 3), to three decimal places.
ANS: x = -12.091, x = 10.091 - Carbon-14 Dating: A 2,500-year-old estimate of Carbon-14 shows a decrease from 20 grams to 12 grams, so 60% remains. P(t) = 20e-0.00012t.
ANS: t = 4256.8 years - Projectile Motion: 72 feet per second, h(t) = -16t2 + 72t.
Work: x = -b/2a ⇒ x = -72/-32.
ANS: Max of 81 ft - Alcohol Consumption: A(x) = -0.018x3 + 1.09x. Are they drunk after 5 hours?
ANS: A(5) = -0.018(5)3 + 1.09(5) = 3.2. Yes, they are drunk. - Rate of Change: Since 2007, y = 0.0238x + 13.4157. What is the rate of change?
ANS: Slope = 0.0238 or 119/5000 - Exponential Growth: 4.8% growth rate since 2008, starting population 350. Find the population in 2012.
ANS: Equation 350e0.048t, P(4) = 424 - Savings Account: $2,500 savings account, interest grows continuously. How long until it reaches $3,300?
Work: 3300 = 2500e0.023t ⇒ 1.32 = e0.023t ⇒ ln(1.32) = 0.023t.
ANS: t = 21.4 years - Logarithmic Equation: log2(x – 3) + log2(x – 5) = 3.
Work: log2(x2 – 5x – 3x + 15) = 3. Multiply both sides by 2 (base conversion) ⇒ x2 – 8x + 15 = 8 ⇒ x2 – 8x + 7 = 0 ⇒ (x – 7)(x – 1).
ANS: x = 7 - Difference Quotient: Find the difference quotient of f(x) = x2 – 6x + 8.
Work: f(x) = [(x + h)2 – 6(x + h) + 8 – (x2 – 6x + 8)] / h. Cancel out terms.
ANS: 2x + h – 6 - Inequality: (x – 6) / (x + 3) ≥ 6.
ANS: [6, ∞)
Mathematical Problem Set 3
- Profit Function: P(x) = -4x2 + 632x – 4000, where x > 10 and x is the number of units produced. Find the total production for $800 profit.
Work: 0 = -4x2 + 632x – 4800. Plug into the quadratic formula.
ANS: x = 150 - Relative Maximum: F(x) = ln(2x – 8.4) – 0.25(x + 4)2 + 9.83.
ANS: 10.2 at x = 5.5 - Real Zeros: 3 – 2 log5 √(x2 + 2x + 3), to three decimal places.
ANS: x = -12.091, x = 10.091 - Carbon-14 Dating: A 2,500-year-old estimate of Carbon-14 shows a decrease from 20 grams to 12 grams, so 60% remains. P(t) = 20e-0.00012t.
ANS: t = 4256.8 years - Projectile Motion: 72 feet per second, h(t) = -16t2 + 72t.
Work: x = -b/2a ⇒ x = -72/-32.
ANS: Max of 81 ft - Alcohol Consumption: A(x) = -0.018x3 + 1.09x. Are they drunk after 5 hours?
ANS: A(5) = -0.018(5)3 + 1.09(5) = 3.2. Yes, they are drunk. - Rate of Change: Since 2007, y = 0.0238x + 13.4157. What is the rate of change?
ANS: Slope = 0.0238 or 119/5000 - Exponential Growth: 4.8% growth rate since 2008, starting population 350. Find the population in 2012.
ANS: Equation 350e0.048t, P(4) = 424 - Savings Account: $2,500 savings account, interest grows continuously. How long until it reaches $3,300?
Work: 3300 = 2500e0.023t ⇒ 1.32 = e0.023t ⇒ ln(1.32) = 0.023t.
ANS: t = 21.4 years - Logarithmic Equation: log2(x – 3) + log2(x – 5) = 3.
Work: log2(x2 – 5x – 3x + 15) = 3. Multiply both sides by 2 (base conversion) ⇒ x2 – 8x + 15 = 8 ⇒ x2 – 8x + 7 = 0 ⇒ (x – 7)(x – 1).
ANS: x = 7 - Difference Quotient: Find the difference quotient of f(x) = x2 – 6x + 8.
Work: f(x) = [(x + h)2 – 6(x + h) + 8 – (x2 – 6x + 8)] / h. Cancel out terms.
ANS: 2x + h – 6 - Inequality: (x – 6) / (x + 3) ≥ 6.
ANS: [6, ∞)
