A wire would enclose an area of 1936 m2, if it is bent into a square. The wire is cut into two pieces. The longer piece is thrice as long as the shorter piece. The long and the short pieces are bent into a square and a circle, respectively. Which of the following choices is closest to the sum of the areas enclosed by the two pieces in square meters?
1243
Wire Problem: Calculating Total Enclosed Area
This problem involves understanding how the length of a wire relates to the perimeter or circumference of the shapes it forms. We will determine the total length of the wire, divide it as specified, and then find the sum of the areas of the resulting square and circle.
1. Determining the Initial Wire Length
The problem states that if the wire is bent into a square, it encloses an area of 1936 m2. We can use this information to find the total length of the wire.
- Let the side of the initial square be \(s\).
- The area of a square is given by the formula: Area = \(s^2\).
- Given Area = 1936 m2.
- So, \(s^2 = 1936\) m2.
- To find the side \(s\), we take the square root of 1936:
- \(s = \sqrt{1936} = 44\) meters.
- The total length of the wire is equal to the perimeter of this initial square.
- The perimeter of a square = \(4 \times s\).
- Total wire length = \(4 \times 44 = 176\) meters.
2. Dividing the Wire into Two Pieces
The total wire length of 176 meters is cut into two pieces. The longer piece is thrice as long as the shorter piece.
- Let the length of the shorter piece be \(x\) meters.
- Then, the length of the longer piece is \(3x\) meters.
- The sum of the lengths of the two pieces equals the total wire length:
- \(x + 3x = 176\)
- \(4x = 176\)
- Now, we solve for \(x\):
- \(x = \frac{176}{4} = 44\) meters.
- So, the shorter piece has a length of 44 meters.
- And the longer piece has a length of \(3 \times 44 = 132\) meters.
| Piece | Length (m) |
|---|---|
| Shorter Piece | 44 |
| Longer Piece | 132 |
| Total Wire Length | 176 |
3. Calculating Area Enclosed by the Square
The longer piece (132 meters) is bent into a square. We need to find the area of this square.
- The perimeter of this new square is 132 meters.
- Let the side of this square be \(s_1\).
- The perimeter of a square = \(4 \times s_1\).
- \(4 \times s_1 = 132\)
- \(s_1 = \frac{132}{4} = 33\) meters.
- The area of this square is \(s_1^2\).
- Area of square = \(33^2 = 1089\) m2.
4. Calculating Area Enclosed by the Circle
The shorter piece (44 meters) is bent into a circle. We need to find the area of this circle.
- The circumference of this circle is 44 meters.
- The formula for the circumference of a circle is \(C = 2\pi r\), where \(r\) is the radius.
- \(2\pi r = 44\)
- Using the approximation \(\pi = \frac{22}{7}\):
- \(2 \times \frac{22}{7} \times r = 44\)
- \(\frac{44}{7} \times r = 44\)
- \(r = \frac{44 \times 7}{44} = 7\) meters.
- The area of a circle is given by the formula: Area = \(\pi r^2\).
- Area of circle = \(\frac{22}{7} \times 7^2\)
- Area of circle = \(\frac{22}{7} \times 49\)
- Area of circle = \(22 \times 7 = 154\) m2.
5. Sum of Enclosed Areas
Finally, we need to find the sum of the areas enclosed by the square and the circle.
- Sum of Areas = Area of Square + Area of Circle
- Sum of Areas = \(1089 \text{ m}^2 + 154 \text{ m}^2\)
- Sum of Areas = \(1243 \text{ m}^2\).
The sum of the areas enclosed by the two pieces is 1243 square meters.
Important Questions from Numerical Computation
Ratio of the angles of a triangle are 1 : 2 : 6. What is the smallest angle?
It would take one machine 4 hours to complete a production order and another machine 2 hour to complete the same order. If both machines work simultaneously at their respective constant rates, the time taken to complete the same order is ________ hours.
Two design consultants, P and Q, started working from 8 AM for a client. The client budgeted a total of USD 3000 for the consultants. P stopped working when the hour hand moved by 210 degrees on the clock. Q stopped working when the hour hand moved by 240 degrees. P took two tea breaks of 15 minutes each during her shift, but took no lunch break. Q took only one lunch break for 20 minutes, but no tea breaks. The market rate for consultants is USD 200 per hour and breaks are not paid. After paying the consultants, the client shall have USD_remaining in the budget.
What is the value of \(1 + \frac{1}{4} + \frac{1}{{16}} + \frac{1}{{64}} + \frac{1}{{256}} + \ldots ?\)
A 1.5 m tall person is standing at a distance of 3 m from a lamp post. The light from the lamp at the top of the post casts her shadow. The length of the shadow is twice her height. What is the height of the lamp post in meters?
