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Problem 19 Easy

A container has a 1\times 1 square bottom, a 3\times 3 open square top, and four congruent trapezoidal sides, as shown. Starting when the container is empty, a hose that runs water at a constant rate takes 35 minutes to fill the container up to the midline of the trapezoids.

How many more minutes will it take to fill the remainder of the container?

  • A.

    70

  • B.

    85

  • C.

    90

  • D.

    95

  • E.

    105

Answer:D
Solution

Extend the lines so the sides become triangles.

Let h be the height of half the container; then the extended cone also has height h since the side lengths of the squares are 1, 2, 3 respectively.

\frac{V_{\text{remained}}}{V_{\text{filled}}} = \frac{\frac{1}{3} \cdot 2h \cdot 2^2 - \frac{1}{3} \cdot 2h \cdot 1^2}{\frac{1}{3} \cdot 2h \cdot 1^2 - \frac{1}{3} \cdot h \cdot 1^2} = \frac{27-8}{8-1} = \frac{19}{7}

Therefore, the time = 35 \times \frac{19}{7} = 95 mins.

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