Question

the area of a rectangle is 24 square centimeters. the length (in centimeters) of the rectangle is given by the equation (x^{2}+3x - 40 = 0). what is the length of the rectangle?
a. 3 cm
b. 5 cm
c. 8 cm
d. 12 cm

Explanation:

1. Recall the area formula for a rectangle:

• The area formula of a rectangle is \(A = lw\), where \(A\) is the area, \(l\) is the length, and \(w\) is the width. Here, assume the length \(l=x\) and we are not given the width - related information clearly in the problem statement text you provided. But if we assume the length \(x\) satisfies some equation (which is not fully visible in your image, let's assume we know the area \(A = 24\) square centimeters).
• If we assume the length \(x\) is a factor of the area. We know that \(A=24\) and we check the options one - by - one.

1. Check each option:

• Option A: If \(l = 3\) cm, and assume \(A=lw = 24\) square centimeters, then \(w=\frac{A}{l}=\frac{24}{3}=8\) cm.
• Option B: If \(l = 5\) cm, then \(w=\frac{24}{5}=4.8\) cm.
• Option C: If \(l = 8\) cm, then \(w=\frac{24}{8}=3\) cm.
• Option D: If \(l = 12\) cm, then \(w=\frac{24}{12}=2\) cm.
• Since the problem seems to be incomplete in the given image, if we assume we are looking for a whole - number factor of 24 as the length (a common assumption in basic geometry problems about rectangles), and if we assume the length and width are whole numbers, and the area is 24 square centimeters, the factors of 24 are \(1,2,3,4,6,8,12,24\). Among the given options, all of them can be lengths of the rectangle when paired with appropriate widths (\(8\) for \(l = 3\), \(4.8\) for \(l = 5\), \(3\) for \(l = 8\), \(2\) for \(l = 12\)). But if we assume we want whole - number side lengths for a basic rectangle problem, and we know that \(24=3\times8=8\times3 = 12\times2=2\times12\).
• If we assume a more common - sense approach where we consider the pairs of whole - number factors of 24, and we have no other information, we can note that if the area \(A = 24\) and we assume the length \(l\) and width \(w\) are non - negative real numbers. If we assume we are looking for a whole - number length, and we consider the pairs \((l,w)\) such that \(lw = 24\), the possible pairs are \((1,24),(2,12),(3,8),(4,6),(6,4),(8,3),(12,2),(24,1)\).
• Among the options, if we assume we are looking for a whole - number length and no other constraints, all options can be lengths of rectangles with area 24 square centimeters when paired with the right width. But if we assume a more "basic" or "common" rectangle situation where we consider whole - number side lengths and we know \(24=3\times8\) or \(24 = 8\times3\) etc., and we assume we are choosing from the given options, if we consider the pairs of factors of 24, when \(l = 8\) cm and \(w = 3\) cm, the area \(A=lw=8\times3 = 24\) square centimeters.

Answer:

C. 8 cm