- 1. 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.

A.) Rs.125,400

B.) Rs.140,800

C.) Rs.235,400

D.) None of these

Answer: Option 'A'

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.

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 = 50âˆš2 ft.

Therefore, inner diameter of the circular road = 50âˆš2.

Hence, inner radius of the circular road = 25âˆš2 ft.

Then, outer radius = 25âˆš2 + 722 = 32âˆš2

The area of the circular road

= Ï€ r_{o}^{2} - Ï€ r_{i}^{2}, where ro is the outer radius and r_{i} is the inner radius.

= 22/7 Ã— {(32 âˆš2)^{2} â€“ (25âˆš2)^{2}}

= 22/7 Ã— 2 Ã—(32 + 25)Ã—(32 â€“ 25)

= 2508 sq. ft.

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

Cost of constructing 50% of the road = 50% of the total cost = 250800/2 = Rs.1,25,400

- 2. A solid metal cylinder of 10 cm height and 14 cm diameter is melted and re-cast into two cones in the proportion of 3 : 4 (volume), keeping the height 10 cm. What would be the percentage change in the flat surface area before and after?

A.) 9%

B.) 15%

C.) 25%

D.) 50%

Answer: Option 'D'

Let radius of cylinder, cone 1, cone 2 = R, r1, r2 respectively.

Let their volumes be V, v1, v2 respectively.

Cylinder is melted and recast into cone 1 & cone 2

So V = v1 + v2.

Ratio of volumes of the two cones v1 : v2 is 3 : 4.

If v1 is 3x, v2 will be 4x

Hence, volume of cylinder V= 7x

The volumes of the cylinder, cone 1 and cone 2 are Ï€R^{2}h, 1/3 Ï€r_{1}^{2}h and 1/3 Ï€r_{2}^{2}h

We know the ratio of the volumes V : v1, v2 is 7 : 3 : 4

So, Ï€R^{2}h : 1/3 Ï€r_{1}^{2}h : 1/3 Ï€r_{2}^{2}h is 7 : 3 : 4

Cancelling Ï€ and h, which are common to all terms, we get R^{2} : 1/3 r_{1}^{2} : 1/3 r_{2}^{2} = 7 : 3 : 4

Or R2 : r_{1}^{2} : r_{2}^{2} = 7 : 3 : 4.

So, if R^{2} is 7k, r12 will be 9k and r22 will be 12k.

Flat surface area of cylinder (sum of the areas of the two circles at the top and bottom of the cylinder) = 2 * Ï€ * R^{2}

Flat surface area of cone 1 & 2 are: Ï€ * r12 & Ï€ * r22 respectively (areas of the circle at the bottom of each of the cones).

Ratio of the flat surface area of cylinder to that of the two cones is 2 * Ï€ * R^{2} : (Ï€ * r12 + Ï€ * r22)

Cancelling Ï€ on both sides of the ratio we get 2R^{2} : (r_{1}^{2} + r_{2}^{2}
Or 14k : 21k

- 3. If x units are added to the length of the radius of a circle, what is the number of units by which the circumference of the circle is increased?

A.) 2 π

B.) 2 π x

C.) x

D.) x^{2}

Answer: Option 'B'

Let the radius of the circle be 'r' units.

The circumference of the circle will therefore be 2 π r units.

If the radius is increased by 'x' units, the new radius will be (r + x) units.

The new circumference will be 2 π(r + x) = 2 πr + 2 πx

Or the circumference increases by 2πx units.

- 4. A square sheet of paper is converted into a cylinder by rolling it along its length. What is the ratio of the base radius to the side of the square?

A.) 1/2π

B.) √2/π

C.) 1/(√2π)

D.) 1/π

Answer: Option 'D'

Surface area of the cylinder = Surface area of the square 2πh = a^{2}

Here height 'h' of the cylinder = side of the square since it is rolled along its length.2πa = a^{2}

Base radius r = a/2π

Ratio of base radius to side of square = a/2π : a = 1/2π

- 5. What is the are of an equilateral triangle of side 16 cm?

A.) 48√3 cm^{2}

B.) 128√3 cm^{2}

C.) 9.6√3 cm^{2}

D.) 64√3 cm^{2}

Answer: Option 'D'

64√3 ^{2}

Area of an equilateral triangle = √3/4 S^{2}

If S = 16, Area of triangle = √3/4 * 16 * 16 = 64√3 cm^{2};

- 6. The perimeter of a triangle is 28 cm and the inradius of the triangle is 2.5 cm. What is the area of the triangle?

A.) 45 cm^{2}

B.) 35 cm^{2}

C.) 25 cm^{2}

D.) 55 cm^{2}

Answer: Option 'B'

Area of a triangle = r × s

Where r is the in radius and s is the semi perimeter of the triangle.

Area of triangle = 2.5 × 28/2 = 35 cm^{2}

- 7. Find the area of a parallelogram with base 24 cm and height 16 cm.

A.) 544 cm^{2}

B.) 484 cm^{2}

C.) 250 cm^{2}

D.) 384 cm^{2}

Answer: Option 'D'

Area of a parallelogram = base × height = 24 × 16 = 384 cm^{2}

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