A juice can is in the shape of a cylinder. It has a height of 122mm and a diameter of 52 mm. What is the longest straw that can fit completely inside the juice can diagonally? Round the answer to the nearest tenth.
To find the longest straw that can fit completely inside the juice can diagonally, you must consider the can as a cylinder and calculate the diagonal of the cylinder's circular base in conjunction with the height of the cylinder. This diagonal represents the longest straight line you could draw within the can, from the bottom of one side to the top of the other, which would be the hypotenuse of a right-angled triangle.
The juice can has a height (h) of 122 mm and a diameter (d) of 52 mm, which means the radius (r) of the base is half the diameter, so r = d/2 = 52 mm / 2 = 26 mm.
The diagonal of the base (the diameter) and the height form a right-angled triangle where:
- The diameter is one leg of the triangle
- The height is the other leg
- The longest straw is the hypotenuse
Let's call the longest straw "L." Using the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b):
c² = a² + b²
In this case:
L² = r² + h²
Since the longest straw would go corner to corner across the circular base, a is the radius of the base (r), and b is the height (h), we plug in the values we have:
L² = (26 mm)² + (122 mm)²
Now, we calculate each squared value:
L² = 676 mm² + 14884 mm²
L² = 15560 mm²
Taking the square root of both sides to solve for L gives us the length of the straw:
L = √15560 mm²
Let's calculate the square root:
L ≈ √15560
L ≈ 124.74 mm
Rounded to the nearest tenth, the longest straw that can fit completely inside the juice can diagonally is approximately 124.7 mm.