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XAT 2015 Quant & DI Q6 of 33: Mensuration

This aptitude question appeared in the quantitative reasoning and data interpretation section of the XAT 2015. A total of 33 questions appeared in this section in XAT 2015. This one is an easy to medium level difficulty question and tests the concept of finding the area of a circular ring.

Question

A circular road is constructed outside a square field. The perimeter of the square field is 200 ft. If the width of the road is 7√2 ft. and cost of construction is Rs.100 per sq. ft., find the lowest possible cost to construct 50% of the total road.

  1. Rs.70,400
  2. Rs.125,400
  3. Rs.140,800
  4. Rs.235,400
  5. None of the above

  Correct Answer      Choice B. The lowest possible cost to construct 50% of the total road is Rs.125,400.


Explanatory Answer

  • What is to be found?

    We have to find the lowest cost to construct 50% of the total road.

    Data Given

    A circular road is constructed outside a square field. So, the road is in the shape of a circular ring.

    If we have to determine the lowest cost of constructing the road, we have to select the smallest circle that can be constructed outside the square.

    Therefore, the inner circle of the ring should circumscribe the square.

    Perimeter of the square = 200 ft.

    Therefore, side of the square field = 50 ft.

  • Compute the inner and outer radii of the circular ring

    The diagonal of the square field is the diameter of the circle that circumscribes it.
    Measure of the diagonal of the square of side 50 ft = 50\\sqrt2\\) ft.
    Therefore, inner diameter of the circular road = 50\\sqrt2\\).
    Hence, inner radius of the circular road = 25\\sqrt2\\) ft.
    Then, outer radius = 25\\sqrt2\\) + 7\\sqrt2\\) = 32\\sqrt2\\)

  • Compute the area of the circular ring

    The area of the circular road
    = π ro2 - π ri2, where ro is the outer radius and ri is the inner radius.

    = \\frac{22}{7}\\) × {(32 \\sqrt2\\))2 – (25\\sqrt2\\))2}
    = \\frac{22}{7}\\)×2×(32 + 25)×(32 – 25)
    = 2508 sq. ft.

    If per sq. ft. cost is Rs. 100, then cost of constructing the road = 2508 × 100 = Rs.2,50,800.

    Cost of constructing 50% of the road = 50% of the total cost = \\frac{250800}{2}\\) = Rs.1,25,400

      The correct answer is Choice B.

Video Explanation

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