the area of the conference table in mr. nathans office must be no more than 175 ft². if the length of the…

Answer:

B. (0 < x\leq7)

Explanation:

Step1: Set up the area formula

The length of the table is (x + 18) (since length is 18 ft more than the width (x)), and the area (A=length\times width), so (A=(x + 18)x). We know that (A\leq175), so ((x + 18)x\leq175).

Step2: Expand the inequality

Expand ((x + 18)x) to get (x^{2}+18x\leq175), or (x^{2}+18x - 175\leq0).

Step3: Solve the quadratic - equation

First, solve the quadratic equation (x^{2}+18x - 175 = 0). The quadratic formula for (ax^{2}+bx + c=0) is (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}). Here, (a = 1), (b = 18), and (c=-175). Then (x=\frac{-18\pm\sqrt{18^{2}-4\times1\times(-175)}}{2\times1}=\frac{-18\pm\sqrt{324 + 700}}{2}=\frac{-18\pm\sqrt{1024}}{2}=\frac{-18\pm32}{2}). The two solutions are (x_1=\frac{-18 + 32}{2}=\frac{14}{2}=7) and (x_2=\frac{-18-32}{2}=\frac{-50}{2}=-25).

Step4: Determine the solution of the inequality

The quadratic function (y=x^{2}+18x - 175) is a parabola opening upwards (because (a = 1>0)). The inequality (x^{2}+18x - 175\leq0) is satisfied when (-25\leq x\leq7). But since (x) represents the width of a table, (x>0). So the possible values of (x) are (0 < x\leq7).