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A scientist is analyzing a cylindrical water tank with a height of 10 units and a radius of 3 units. Calculate the ratio of the volume of this cylinder to the volume of a cone with the same base and height.
The ratio of the volume of the cylinder to the volume of the cone is:
Solution: To find the shortest altitude, we first compute the area using Herons formula. The semi-perimeter is:
Question: A historian discovers an ancient right triangle diagram where the hypotenuse is $ x $, and the radius of the inscribed circle is $ r $. What is the ratio of the area of the circle to the area of the triangle?
The ratio of the circles area to the triangles area is:
Question: An epidemiologist models a region as a rectangle measuring 5 cm by 12 cm, inscribed in a circular boundary. What is the circumference of the circle?
Solution: The diagonal of the rectangle is the diameter of the circle. Using the Pythagorean theorem:
Question: A triangle has side lengths of 9 cm, 12 cm, and 15 cm. What is the length of the longest altitude?
The longest altitude corresponds to the shortest side (9 cm):
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