Find the volume of a prism 20cm long and whose cross-section is an equilateral triangle of side 9cm.

Here are the solutions to your questions. 11. Find the volume of a prism 20cm long and whose cross-section is an equilateral triangle of side 9cm. Step 1: Calculate the area of the equilateral triangle cross-section. The formula for the area of an equilateral triangle with side length a is A = sqrt(3)4a^2. Given a = 9 cm. A = sqrt(3)4(9 cm)^2 A = sqrt(3)4(81 cm^2) A = 81sqrt(3)4 cm^2 Step 2: Calculate the volume of the prism. The formula for the volume of a prism is V = Area of cross-section × length. Given length = 20 cm. V = 81sqrt(3)4 cm^2 × 20 cm V = 81sqrt(3) × 5 cm^3 V = 405sqrt(3) cm^3 Using sqrt(3) ≈ 1.732: V ≈ 405 × 1.732 cm^3 V ≈ 701.46 cm^3 The volume of the prism is approximately 701.46 cm^3 (or exactly 405sqrt(3) cm^3). 12. The equation of a line L1 is 4y - 3x + 8 = 0. Find the equation of line L2 which is perpendicular to L1 and passes through (2, 7). Step 1: Find the slope of line L1. Rearrange the equation 4y - 3x + 8 = 0 into the slope-intercept form y = mx + c. 4y = 3x - 8 y = (3)/(4)x - 2 The slope of L1 is m_1 = (3)/(4). Step 2: Find the slope of line L2. Since L2 is perpendicular to L1, its slope m_2 is the negative reciprocal of m_1. m_2 = -(1)/(m_1) m_2 = -(1)/(3/4) m_2 = -(4)/(3) Step 3: Find the equation of line L2. Use the point-slope form y - y_1 = m(x - x_1) with m_2 = -(4)/(3) and the point (x_1, y_1) = (2, 7). y - 7 = -(4)/(3)(x - 2) Multiply both sides by 3 to eliminate the fraction: 3(y - 7) = -4(x - 2) 3y - 21 = -4x + 8 Rearrange into the general form Ax + By + C = 0: 4x + 3y - 21 - 8 = 0 4x + 3y - 29 = 0 The equation of line L2 is 4x + 3y - 29 = 0. 13. Solve for x in 3^2x+1 + 5(3^x) - 2 = 0. Step 1: Rewrite the exponential term using exponent rules. 3^2x+1 = 3^2x · 3^1 = (3^x)^2 · 3 Substitute this back into the equation: 3(3^x)^2 + 5(3^x) - 2 = 0 Step 2: Make a substitution to form a quadratic equation. Let y = 3^x. Since 3^x is always positive, y > 0. 3y^2 + 5y - 2 = 0 Step 3: Solve the quadratic equation for y. Factor the quadratic equation: (3y - 1)(y + 2) = 0 This gives two possible solutions for y: 3y - 1 = 0 3y = 1 y = (1)/(3) y + 2 = 0 y = -2 Step 4: Substitute back y = 3^x and solve for x. Since y must be positive, we discard y = -2. Using y = (1)/(3): 3^x = (1)/(3) 3^x = 3^-1 Therefore, x = -1 The solution for x is -1. 14. Express 2-sqrt(3)2+sqrt(3) + 4-sqrt(3)2-sqrt(3) in the form a+bsqrt(c). Step 1: Find a common denominator for the two fractions. The common denominator is (2+sqrt(3))(2-sqrt(3)). Using the difference of squares formula (A+B)(A-B) = A^2 - B^2: (2+sqrt(3))(2-sqrt(3)) = 2^2 - (sqrt(3))^2 = 4 - 3 = 1 Step 2: Rewrite each fraction with the common denominator and combine. 2-sqrt(3)2+sqrt(3) + 4-sqrt(3)2-sqrt(3) = (2-sqrt(3))(2-sqrt(3))(2+sqrt(3))(2-sqrt(3)) + (4-sqrt(3))(2+sqrt(3))(2-sqrt(3))(2+sqrt(3)) Since the common denominator is 1, the expression simplifies to the sum of the new numerators: = (2-sqrt(3))^2 + (4-sqrt(3))(2+sqrt(3)) Step 3: Expand and simplify each term in the numerator. For the first term, (2-sqrt(3))^2: (2-sqrt(3))^2 = 2^2 - 2(2)(sqrt(3)) + (sqrt(3))^2 = 4 - 4sqrt(3) + 3 = 7 - 4sqrt(3) For the second term, (4-sqrt(3))(2+sqrt(3)): (4-sqrt(3))(2+sqrt(3)) = 4(2) + 4(sqrt(3)) - sqrt(3)(2) - (sqrt(3))(sqrt(3)) = 8 + 4sqrt(3) - 2sqrt(3) - 3 = 5 + 2sqrt(3) Step 4: Add the simplified terms. (7 - 4sqrt(3)) + (5 + 2sqrt(3)) = 7 + 5 - 4sqrt(3) + 2sqrt(3) = 12 - 2sqrt(3) The expression in the form a+bsqrt(c) is 12 - 2sqrt(3). 3 done, 2 left today. You're making progress.