A sand timer tells us how much time we have to answer a trivia question. We are given the height of the sand. From the example, we know the dimensions of the timer and the rate at which the sand falls. We want to know how much time we have to answer the question. Let's make a diagram of the timer from the example and the timer from our situation.
First, we need to determine the unknown radius. We can think of the middle part of our timer as a two dimensional slice. Then, we get two triangles instead of two cones. Let's name the unknown variable x.
Next, we need to use our knowledge about similar triangles. We can see that ∠ A and ∠ D are both right angles. We can also see that ∠ C and ∠ F are halves of identical angles. Therefore, they are congruent. By the triangle similarity theorems, this means that the triangles are similar and their corresponding sides are proportional.
DE/AB=DF/AC
Let's substitute x for DE, 10 for AB, 30 for DF, and 24 for AC into the equation and solve for x.
The radius of our triangle is 12.5. Let's go back to our three dimensional timers and add the value of the radius.
Now, we can find the time it takes for the sand to fall in the same way as in the example. Let's substitute 12.5 for r and 30 for h into the formula for the volume of a cone to find the volume of sand in the timer.
Finally, we know that the sand falls at a rate of 50 cubic millimeters per second. We can use this rate to determine how much time we have to answer the question.
1562.5π mm^3 * 1sec/50 mm^3 ≈ 98.17sec
We have about 98.17 seconds to answer the question.
