A piece of wire, which is 120 centimetres, was divided into two parts with a ratio of 11 to 4. A circle was shaped from the long part and the square was shaped from the short one. Determine the ratio between the area of the square and that of the circle in its simplest form, 𝜋 equals 22 over seven.
So let’s start looking at the information in the question. We’re told that a piece of wire of length, 120 centimetres, is divided in the ratio of 11 to 4. The long part of the wire is made into a circle. And the short part is made into a square. We will need to find the area of the circle and the area of the square. So let’s start by working on the actual lengths that our wire was split into. When the wire of 120 centimetres was split into the ratio 11 to 4, this means that it was split into 15 parts, since that’s 11 plus four.
If we divide 120 by the number of parts 15, this means that each part has value of eight. So if we take our ratio 11 to 4 and multiply both parts of ratio by eight, this will mean that our wire was split in the ratio of 88 centimetres to 32 centimetres. This means that the wire forming the circle is 88 centimetres and the wire forming the square is 32 centimetres. So let’s start our calculations for the circle. We know that the distance of the wire is 88 centimetres. And this corresponds to the circumference of the circle; that is, the distance around the outside. The circumference of a circle can also be calculated using the formula two 𝜋𝑟, where 𝑟 is the radius.
So let’s see if we can work out the radius of our circle. We can start by substituting the known value of the circumference of 88 and we can write this equals two times 22 over seven times 𝑟. We use 22 over seven for 𝜋 since we were told that in the question. We can simplify the right-hand side of our equation by multiplying two by 22 over seven giving us 88 equals 44 over seven times 𝑟. To begin rearranging them to find 𝑟, we can start by multiplying both sides by seven giving us 88 times seven equals 44𝑟. We can then divide both sides of our equation by 44 giving us 88 times seven over 44 equals 𝑟. And since we can see that 44 goes into the numerator and the denominator, this will mean that 𝑟 is equal to two times seven. So 𝑟 is 14 centimetres.
So now, we know our radius. We can work right the area of our circle. And we can use the formal at the area of a circle equals 𝜋𝑟 squared. We can substitute in our values of 22 over seven for 𝜋 and the radius is 14 squared. Since 14 squared is equal to 14 times 14 we can evaluate our area as 22 times 196 over seven. We can simplify this calculation by noticing that seven goes into seven and 196, leaving asked to calculation 22 times 28. So we can evaluate this as the area of our circle is 616 centimetres squared.
So now, let’s look at finding the area of our square. Since the length of the wire is 32 centimetres, this means that the perimeter of our square is 32 centimetres. We know that if we have the perimeter of a square, this will be equivalent to four times the length. So we can substitute the values of 32 for the perimeter into our equation, giving us 32 equals four times the length, giving us the length of our square is eight centimetres. To find the area of a square, we use the formula that the area of a square is equal to the length squared or 𝑙 squared. And since our length is eight, then eight squared will give us 64 centimetres squared.
To finish the question then, we need to find the ratio between the area of the square and that of the circle. When we write our ratio, it’s very important that we write them in the same order as what we’re asked for. In this case, the area of the square will be first and the area of the circle will be second. So we can write our values as 64 to 616. And we don’t include units in a ratio.
We’re asked for ratio in its simplest form. So let’s see if we can simplify our ratio. Since both our values are divisible by four, we can divide both sides by ratio by four, which will give us 16 to 154. We can now notice that both of our numbers are even. So we must be able to divide both sides by two, which will give us eight to 77. And since the only factors of 77 are seven under 11, then we can’t simplify our ratio anymore. So our final answer is the ratio eight to 77.