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Chapter 6: Problem 26

What is the largest possible area for a parallelogram that has pairs of sides with lengths 5 and \(9 ?\)

Short Answer

Expert verified
The largest possible area for a parallelogram that has pairs of sides with lengths \(5\) and \(9\) is \(45\) square units.

Step by step solution

01

Understanding the formula for the area of a parallelogram.

In a parallelogram, the area can be found by multiplying the base and the height. The base can be the length of either pair of parallel sides, and the height is the perpendicular distance between the two parallel sides. The formula for the area of a parallelogram is given by: \[A = base \times height\]
02

Using trigonometry to find the height.

Let's consider one of the angles in the parallelogram, which we'll call \(\theta\). Since we know one side length (\(5\)) and the length of one of the adjacent sides (\(9\)), we can use the sine function to find the height. Using one of the triangles within the parallelogram, we have: \[height = 5\sin{\theta}\] Now we just need to determine what value of \(\theta\) will give us the maximum height.
03

Finding the maximum height.

To find the maximum height, we should consider the fact that the sine function reaches its maximum value of \(1\) at an angle of \(90^\circ\) (or \(\frac{\pi}{2}\) radians). Therefore, to get the maximum height, we need \(\theta\) to be \(90^\circ\). Plugging this value into the height formula, we get: \[height_{max} = 5\sin{(90^\circ)} = 5\]
04

Calculating the maximum area of the parallelogram.

Now that we have the maximum height, we can use the formula for the area of a parallelogram to find the maximum area. The base, in this case, is the length of other pair of parallel sides which is \(9\). Since we have the height now we can plug in these values into the formula: \[A_{max} = base \times height_{max} = 9 \times 5 = 45\] Therefore, the largest possible area for a parallelogram that has pairs of sides with lengths \(5\) and \(9\) is \(45\) square units.

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