Sandy is going on a business trip to Washington state. She has a 46 inch umbrella to pack since she will be there during rainy season. She plans to pack the umbrella in the diagonal of her suitcase, but she has to buy the suitcase first. The airline company will allow a suitcase to be 24 inches long. When Sandy buys the suitcase, how wide to the nearest tenth does the suitcase need to be for the umbrella to fit?
Assuming she buys a 24" suitcase, then
24^2+w^2 = 46^2
w = 39.24
odd; width is usually less than length
Now, if you add a 3rd dimension, so the umbrella can lie between opposite corners of the 3D space, then you have, for height h,
24^2 + w^2 + h^2 = 46^2
To determine the width of the suitcase needed for the umbrella to fit, we can use the Pythagorean theorem, which states that the square of the hypotenuse (the longest side) of a right-angled triangle is equal to the sum of the squares of the other two sides.
In this case, the length of the suitcase represents one of the sides, the width represents the other side, and the diagonal of the suitcase (equal to the length of the umbrella) represents the hypotenuse.
Applying the theorem, we have:
Hypotenuse^2 = Length^2 + Width^2
Since the length of the suitcase is given as 24 inches, we can substitute it into the equation:
Hypotenuse^2 = 24^2 + Width^2
To find the width, we need to rearrange the equation:
Width^2 = Hypotenuse^2 - 24^2
Now we substitute the length of the umbrella, which is 46 inches, into the equation:
Width^2 = 46^2 - 24^2
Solving this equation gives us the value of Width^2.
Width^2 = 2116 - 576
Width^2 = 1540
To find the width, we take the square root of both sides:
Width = √1540
Using a calculator, we find that the width is approximately 39.2 inches (rounded to the nearest tenth).
Therefore, Sandy needs to buy a suitcase with a width of approximately 39.2 inches for the umbrella to fit in the diagonal.