Show that the area of the triangle in the figure above is given by: A(x) = -x2/2 + 4x - 15/2

Welcome back ✰_✰..G-Mathabie!! — missed you this week. Here's the solution to the problem. 23.1) Step 1: Identify the base and height of the right-angled triangle. From the figure, the base BC = 3-x and the height AB = x+2. Step 2: Write the formula for the area of a triangle and substitute the given expressions. The area of a triangle is given by A = (1)/(2) × base × height. A(x) = (1)/(2) (3-x)(x+2) Step 3: Expand and simplify the expression for the area. A(x) = (1)/(2) (3x + 6 - x^2 - 2x) A(x) = (1)/(2) (-x^2 + x + 6) A(x) = -(x^2)/(2) + (x)/(2) + (6)/(2) A(x) = -(x^2)/(2) + (x)/(2) + 3 The problem statement gives A(x) = -(x^2)/(2) + 4x - (15)/(2). There seems to be a discrepancy between the derived area and the given area function. Let's re-check the calculation. Re-evaluating the problem statement for 23.1: "Show that the area of the triangle in the figure above is given by: A(x) = -(x^2)/(2) + 4x - (15)/(2)". The image shows the base as 3-x and the height as x+2. Let's assume the given area function in the question is correct and the diagram might be misleading or there's a different interpretation of the sides. However, based on the diagram, the area is A(x) = -(x^2)/(2) + (x)/(2) + 3. If we are to show that the area is A(x) = -(x^2)/(2) + 4x - (15)/(2), then the sides of the triangle in the diagram must be different from what is labeled. Let's assume the question implies that the area is A(x) = -(x^2)/(2) + 4x - (15)/(2) and we need to work with this function for the subsequent parts, even if it doesn't directly match the diagram's dimensions. This is a common type of question where the diagram is illustrative but the function provided is the one to use. Given the instruction "Show that the area... is given by: A(x) = -(x^2)/(2) + 4x - (15)/(2)", and the diagram showing sides x+2 and 3-x, these two pieces of information are contradictory. If I use the diagram, I get A(x) = -(x^2)/(2) + (x)/(2) + 3. If I assume the given formula is correct, then the diagram's labels are incorrect. I will proceed by assuming the question intends for us to use the provided formula A(x) = -(x^2)/(2) + 4x - (15)/(2) for parts 23.2 and 23.3, and that the "Show that" part of 23.1 is either a misstatement or refers to a different set of dimensions not explicitly shown or derivable from the current diagram. Since I cannot derive the given formula from the diagram, I will state this discrepancy and then use the given formula for the subsequent parts. Discrepancy Note: Based on the provided diagram with base 3-x and height x+2, the area is A(x) = (1)/(2)(3-x)(x+2) = -(x^2)/(2) + (x)/(2) + 3. This does not match the formula A(x) = -(x^2)/(2) + 4x - (15)/(2) given in the question for 23.1. I will proceed with the given formula A(x) = -(x^2)/(2) + 4x - (15)/(2) for the following calculations, assuming it is the intended function for the problem. 23.2) Step 1: To find the value of x for which the area is maximum, we need to find the derivative of A(x) with respect to x and set it to zero. Given the area function: A(x) = -(x^2)/(2) + 4x - (15)/(2) Step 2: Differentiate A(x) with respect to x. A'(x) = (d)/(dx) ( -(x^2)/(2) + 4x - (15)/(2) ) A'(x) = -(2x)/(2) + 4 - 0 A'(x) = -x + 4 Step 3: Set the derivative to zero to find the critical point. -x + 4 = 0 -x = -4 x = 4 To confirm this is a maximum, we can check the second derivative: A''(x) = -1, which is less than 0, indicating a maximum. The value of x for which the area will be maximum is x=4. 23.3) Step 1: Substitute the value of x found in 23.2 into the area function A(x) to calculate the maximum area. A(x) = -(x^2)/(2) + 4x - (15)/(2) Substitute x=4: A(4) = -((4)^2)/(2) + 4(4) - (15)/(2) Step 2: Simplify the expression to find the maximum area. A(4) = -(16)/(2) + 16 - (15)/(2) A(4) = -8 + 16 - 7.5 A(4) = 8 - 7.5 A(4) = 0.5 The maximum area of the given triangle is 0.5 square units. Drop the next question 📸