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Answer:
Step-by-step explanation:
Exercise 2: The landscaper's problem
A square plot ABCD has a side length of 10 m. A landscaper wants to plant grass in two parts of this plot, represented by the square AEFG and the triangle BFC. Let's assume AE = x and denote A(x) as the area of the plot with grass (shaded in gray in the diagram).
1. To find the possible values for x, we need to determine the range of x. Since AE is a side of the square AEFG, x must be greater than or equal to 0 and less than or equal to 10. Therefore, the interval for x is [0, 10].
2. To demonstrate that A(x) = x^2 - 5x + 50, we can find the areas of the square and the triangle separately, and then add them together. The area of the square is x^2, and the area of the triangle can be calculated using the formula for the area of a triangle: (base * height) / 2. In this case, the base is 10 - x (since BFC is the remaining part of the square), and the height can be found by drawing a perpendicular from F to the base BC and denoting the foot of this perpendicular as H. By applying the Pythagorean theorem to the right triangle BHF, we find that FH = sqrt((10 - x)^2 - 5^2). Therefore, the area of the triangle is (10 - x) * sqrt((10 - x)^2 - 5^2) / 2. Adding these two areas together, we get A(x) = x^2 - 5x + 50.
3. The constraint imposed on the landscaper is that the area of the plot with grass should be equal to 74 m^2. To solve this, we set A(x) equal to 74 and solve for x. This equation can be rewritten as x^2 - 5x + 50 - 74 = 0, which simplifies to x^2 - 5x - 24 = 0. To factorize this quadratic equation, we can look for two numbers whose product is -24 and whose sum is -5. These numbers are -8 and 3. Therefore, we can factorize the equation as (x + 3)(x - 8) = 0.
a. Justifying that solving the equation x^2 - 5x - 24 = 0 is equivalent to finding x values that satisfy the condition A(x) = 74.
b. Expanding the expression (x + 3)(x - 8) to x^2 - 5x - 24.
c. Using the solutions x = -3 and x = 8 to answer the landscaper's problem, indicating that x must be within the interval [0, 10] and choosing the appropriate solution(s) based on this interval.
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