Forgotten formula

The workings

In today’s Divertimento, a student faces a maths exam. Maths tests are often associated with a certain amount of stress and anxiety, but usually do not reach the levels that Professor Igor Tamm (who was awarded the Nobel Prize in Physics in 1958, along with Pavel Cherenkov and Ilya Frank) would be subjected to. His period as a professor at Odessa University coincided with the civil war in Russia that followed the October Revolution of 1917. Famine and looting spread among the inhabitants of Odessa as a result of the clashes in the city. Tamm went out to look for food in a nearby town and was seized by an anti-communist group, who took him for an agitator on the opposing side. When they threatened to shoot him, Tamm claimed that he was not a communist, and that he worked as a mathematics teacher at the university. The leader of the group, incredulous, asked him for an estimate of the error that occurs when truncating the Taylor development of a function in the n-th term. Tamm answered correctly, which earned him his freedom.

More details about the story: Three high-stakes math exams

The fun

A student taking an exam on the applications of Trigonometry is asked for the area of a trapezoid of which the two bases \(B\) and \(b\) are known, which are two integers, and the angles formed by the non-parallel sides with the larger base, 30º and 60º respectively. The student calculates the height \(h\) of the trapezoid, but does not remember the formula for the area of the trapezoid (which is not in the topic being examined) and does not know how to deduce this formula; hesitating between

$$ S_1=\frac{B+b}{2}\cdot h \qquad \text{and} \qquad S_2= \frac{B\cdot b}{2} + h.$$

After calculating both values with the data available and obtaining the same result, the student chooses to write in the exam after the calculation of the height

Therefore, the area of the trapezoid is …

without detailing the formula he has used.

We ask if the student made a mistake in obtaining the same value by applying both formulas or, if not, the relationship that exists between the lengths of the bases for this equality to occur.

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Solution

We encourage the readers to try to solve the divertimento for themselves. Whether you succeed or not, you can always consult the solution in this link.