Video: CBSE Class X • Pack 2 • 2017 • Question 16

CBSE Class X • Pack 2 • 2017 • Question 16

05:25

Video Transcript

Three semicircles each of diameter three centimeters, a circle of diameter 4.5 centimeters, and a semicircle of radius 4.5 centimeters are drawn in the given figure. Find the area of the shaded region.

Looking at our figure, we see the three semicircles of diameter three centimeters. And we can label the area of one of those semicircles 𝐴 sub three. We also can see the circle with a diameter of 4.5 centimeters highlighted in green. We’ll name this area 𝐴 sub two. And finally, there’s the larger semicircle of radius 4.5 centimeters. We’ll call this area 𝐴 sub one. And given all this, we want to solve for the area of the shaded region in the figure. We’ll refer to this area as 𝐴 sub sh.

And by looking at the figure, we can see that that area is equal to 𝐴 sub one, the area of the larger semicircle, minus 𝐴 sub two, the green circle, minus two times 𝐴 sub three, the area of the smaller semicircle, plus 𝐴 sub three. We add this final 𝐴 sub three term onto this expression because of the extension of the shaded region below the line of the larger semicircle. Looking at this expression, we see we can combine the two 𝐴 sub three terms. This simplifies our expression to 𝐴 sub one minus 𝐴 sub two minus 𝐴 sub three. That’s the area of the shaded region.

We’ll now use the information given about each to these areas to solve for 𝐴 sub sh. As we do, we’ll use the fact that the area of a circle is equal to 𝜋 times its radius squared. And that two times the radius of a circle is equal to its diameter. We can now write out the area 𝐴 one in terms of this relationship. That area is equal to 𝜋 times the radius, 4.5 centimeters, squared all divided by two, because 𝐴 one is a semicircle rather than a full circle.

Next, calculating the area 𝐴 two, this is equal to 𝜋 times the diameter of the circle divided by two. That gives us the radius all squared. And finally, the area 𝐴 three is equal to 𝜋 times the diameter of three centimeters divided by two all squared. And all that divided by two because 𝐴 three is a semicircle rather than a full circle. We can now begin to simplify this expression for 𝐴 sub sh. And we’ll start doing it with the factors of two that appear in the denominators of each term.

In the expression for 𝐴 one, that factor of two is already separated out. In the expression for 𝐴 two, one over two quantity squared can be rewritten as one over four. And in the expression for 𝐴 three, we have one over two squared, which will give us one over four. And that itself divided by two, which will give us overall one over eight.

As we consider once more this entire expression, we see that a factor of 𝜋 over two appears in each of the three terms. When we factor 𝜋 over two out from each term, we’re left with the resulting expression in our brackets. We see that this expression involves 4.5 centimeters quantity squared minus 4.5 centimeters quantity squared over two. These two terms combine to equal 4.5 centimeters quantity squared over two, leaving us with just two terms in brackets. We see that now we can factor out one over two from each of the terms. We can now start to calculate each of these remaining terms.

We’ll start with three centimeters quantity squared over two. This is equal to nine centimeters squared over two or 4.5 square centimeters. When we calculate the area of the first term, 4.5 centimeters quantity squared, multiplying 4.5 times itself, we find a result of 20.25. 20.25 square centimeters minus 4.5 square centimeters is equal to 15.75 square centimeters. If we want to calculate an exact answer for this solution, we can notice that 15.75 can be rewritten as a fraction. 63 divided by four is equal to 15.75. So when we substitute this exact fraction in for our decimal and multiply through by 𝜋 over four, we find that the area of the shaded region is exactly 63𝜋 over 16 square centimeters.

If we want to calculate a decimal approximation of this area though, we can assume a specific truncated value for 𝜋. We can assume that 𝜋 is exactly equal to 3.14. To calculate then this approximate area of the shaded region, we’ll multiply 63 by this value for 𝜋. The result of multiplying these values together is 197.82. So now, as a final step, we’ll divide this result by 16.

Setting up this long division, we find that 16 goes into 19 one time. 16 divides into 37 two times. It divides into 58 three times. It divides into 102 six times. And it divides into 60 three times. Since we approximated 𝜋 to two decimal places, we’ll only keep two decimal places in this result of our long division. This tells us that 𝐴 sub sh is approximately 12.36 centimeters squared. That’s the area of the shaded region in the figure.

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