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2017-11-27 Intermediate


What percentage of the square's area is colored blue?

15 can be written as the difference between two positive perfect squares: \[15 = 16-1= 4^2 - 1^2.\] Which of the following numbers cannot be written as the difference of two positive perfect squares?

A factory manufactures wooden squares whose side length should be one meter. However, the side length of the squares is not always \(\SI{1}{\meter}\) but varies uniformly between \(\SI{1}{\meter}\) and \(\SI{1.1}{\meter}.\) Note that the height and the breadth of each square piece remain equal to each other.

Two inspectors, Alice and Bob, want to calculate the average area of a square.

  • Alice claims that the average area is \(\SI{1.1025}{\meter\squared}\) because the average side length is \(\SI{1.05}{\meter}\) and \(1.05 \times 1.05 = 1.1025.\)
  • Bob claims that the average area is \(\SI{1.1050}{\meter\squared}\) because the the area is between \((\SI{1}{\meter} \times \SI{1}{\meter})\) and \((\SI{1.1}{\meter} \times \SI{1.1}{\meter}),\) and \(\frac{1 \times 1 + 1.1 \times 1.1}{2} = 1.105\).

Who is correct?

An accelerometer is made using tubular sections, as shown below. The accelerometer is placed in a rocket that travels in the \(\vec{x}\) direction, and the liquid levels are steady at \(\SI{20}{\centi\meter}\) and \(\SI{10}{\centi\meter}.\)

Find the magnitude of the rocket’s acceleration (in \(\text{m/s}^2\)). Submit your answer to 3 decimal places.

Details and Assumptions:

  • The diameter of the tube is much smaller than the heights of the liquid in the two arms.
  • \(g = 10 \text{ m/s}^{2}.\)

A ladder is placed on a stairway (as shown). What is the length of the shorter leg?

Details and Assumptions:

  • Each step has the same horizontal run of \(15 \text{ cm}\) and vertical run of \(10 \text{ cm}.\)
  • Each leg of the ladder rests on the horizontal midpoint of a step.
  • The angles that the legs make with the steps are congruent.

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