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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=161=4212.15 = 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 1 m\SI{1}{\meter} but varies uniformly between 1 m\SI{1}{\meter} and 1.1 m.\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 1.1025 m2\SI{1.1025}{\meter\squared} because the average side length is 1.05 m\SI{1.05}{\meter} and 1.05×1.05=1.1025.1.05 \times 1.05 = 1.1025.
  • Bob claims that the average area is 1.1050 m2\SI{1.1050}{\meter\squared} because the the area is between (1 m×1 m)(\SI{1}{\meter} \times \SI{1}{\meter}) and (1.1 m×1.1 m),(\SI{1.1}{\meter} \times \SI{1.1}{\meter}), and 1×1+1.1×1.12=1.105\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 x\vec{x} direction, and the liquid levels are steady at 20 cm\SI{20}{\centi\meter} and 10 cm.\SI{10}{\centi\meter}.

Find the magnitude of the rocket’s acceleration (in m/s2\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 m/s2.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 cm15 \text{ cm} and vertical run of 10 cm.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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