Write the given function for the drone's height.

Here are the solutions to the questions: 21. a) Step 1: Write the given function for the drone's height. h(t) = t^2 - 6t + 8 Step 2: Substitute t = 4 seconds into the function. h(4) = (4)^2 - 6(4) + 8 h(4) = 16 - 24 + 8 h(4) = -8 + 8 h(4) = 0 The height of the drone at t = 4 seconds is 0 units. b) Step 1: To find the minimum height, complete the square for the function h(t) = t^2 - 6t + 8. Take half of the coefficient of t and square it: ((-6)/(2))^2 = (-3)^2 = 9. Step 2: Add and subtract this value to the expression. h(t) = (t^2 - 6t + 9) - 9 + 8 Step 3: Factor the perfect square trinomial and simplify the constants. h(t) = (t - 3)^2 - 1 The minimum value of (t-3)^2 is 0 when t=3. Therefore, the minimum height is -1. The minimum height of the drone is -1 units. 22. a) Step 1: Write the quadratic equation. x^2 - 2x - 8 = 0 Step 2: Find two numbers that multiply to -8 and add to -2. These numbers are -4 and 2. Step 3: Rewrite the middle term using these numbers and factor by grouping. x^2 - 4x + 2x - 8 = 0 x(x - 4) + 2(x - 4) = 0 (x - 4)(x + 2) = 0 Step 4: Set each factor to zero and solve for x. x - 4 = 0 or x + 2 = 0 x = 4 or x = -2 The solutions to the equation are x = 4 and x = -2. b) Step 1: The dimensions of a metal sheet must be positive. From part (a), the solutions are x = 4 and x = -2. Step 2: Discard the negative solution as a dimension cannot be negative. The two possible dimensions are 4 units and 2 units. (Assuming the dimensions are x and x-2 or similar, or simply the values of x that represent dimensions). Given the context of dimensions, we usually take the positive values. If the dimensions are x and x-2, then x=4 gives dimensions 4 and 2. If x=-2, it would give -2 and -4, which are not valid. So, the dimensions are 4 and 2. 23. a) Step 1: Write the expression for the robot's wheel rotation. w = 3x^3y^2 × 4x^5y^4 Step 2: Multiply the numerical coefficients. 3 × 4 = 12 Step 3: Apply the product rule for exponents (a^m × a^n = a^m+n) for the x terms and y terms. For x: x^3 × x^5 = x^3+5 = x^8 For y: y^2 × y^4 = y^2+4 = y^6 Step 4: Combine the results. w = 12x^8y^6 The simplified expression is 12x^8y^6. b) Step 1: Substitute x = 2 and y = 3 into the simplified expression from part (a). w = 12(2)^8(3)^6 Step 2: Calculate the powers. 2^8 = 256 3^6 = 729 Step 3: Multiply the values. w = 12 × 256 × 729 w = 3072 × 729 w = 2239488 The robot's wheel rotation value is 2239488. 24. a) A composite number is a positive integer that has at least one divisor other than 1 and itself. Numbers between 20 and 40: 21 (divisors: 3, 7) 22 (divisors: 2, 11) 24 (divisors: 2, 3, 4, 6, 8, 12) 25 (divisors: 5) 26 (divisors: 2, 13) 27 (divisors: 3, 9) 28 (divisors: 2, 4, 7, 14) 30 (divisors: 2, 3, 5, 6, 10, 15) 32 (divisors: 2, 4, 8, 16) 33 (divisors: 3, 11) 34 (divisors: 2, 17) 35 (divisors: 5, 7) 36 (divisors: 2, 3, 4, 6, 9, 12, 18) 38 (divisors: 2, 19) 39 (divisors: 3, 13) Any four composite numbers between 20 and 40 are: 21, 22, 24, 25. (Other valid answers include 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39). b) Step 1: Choose one of the numbers from part (a). Let's choose 24. Step 2: Find its prime factorization. 24 = 2 × 12 12 = 2 × 6 6 = 2 × 3 So, 24 = 2 × 2 × 2 × 3 = 2^3 × 3. The prime factorization of 24 is 2^3 × 3. (If a different number was chosen, the prime factorization would be different, e.g., for 21: 3 × 7; for 22: 2 × 11; for 25: 5^2). 25. a) Step 1: Write the expression for profit. P = 5x^2 - 20x Step 2: Find the greatest common factor (GCF) of the terms 5x^2 and -20x. The GCF of 5 and 20 is 5. The GCF of x^2 and x is x. So, the GCF of 5x^2 and -20x is 5x. Step 3: Factor out the GCF. P = 5x(x - 4) The factorized expression is 5x(x - 4) . b) Step 1: Substitute x = 7 into the profit equation. P = 5x^2 - 20x P = 5(7)^2 - 20(7) Step 2: Calculate the values. P = 5(49) - 140 P = 245 - 140 P = 105 The profit when x = 7 (representing 7 thousand items) is 105 units of profit. (The units of profit are not specified, but it's a numerical value). 3 done, 2 left today. You're making progress.