第10类NCERT解决方案–第12章与圆有关的领域–练习12.1

问题1.两个圆的半径分别为19厘米和9厘米。求出圆的半径，该圆的周长等于两个圆的周长之和。

解决方案：

Since we have to find the radius of the final circle we will use the formula of the circumference of circle C = 2πr.

Radius (r₁) = 19 cm

Radius (r₂) = 9 cm

Circumference of a circle with a radius of 19 cm is 2π * 19 = 38π cm.

Circumference of a circle with a radius of 9 cm is 2π * 9 = 18π cm.

Total Circumference is = 38π + 18π = 56π cm

Now using C = 2πr₃  we will find r₃ which is:

56π = 2π * (r₃)

Therefore, the radius of the circle which has a circumference equal to the sum of the circumference of the given two circles is 28 cm.

问题2.两个圆的半径分别是8厘米和6厘米。求出面积等于两个圆的面积之和的圆的半径。

解决方案：

Area of circle is = πr² cm²

Area of circle₁ = 64π cm²

Area of circle₂ = 36π cm²

Total Area = 100π cm²

Now in order to find the radius of the required circle:

100π = πr²

r = 10 cm.

Therefore, the radius of the circle having an area equal to the sum of the areas of the two circles is 10cm.

问题3.该图描绘了一个射箭目标，从中心向外五个标记区域分别标记为金，红，蓝，黑和白。代表金牌得分的区域的直径为21厘米，其他每个带的宽度为10.5厘米。找出五个得分区域中每个区域的面积。

解决方案：

Radius (r₁) of gold region (i.e., 1st circle) = 21/2 = 10.5 cm

Also, as per the question, it is Given that each circle is 10.5 cm wider than the previous circle.

Therefore, radius (r₂) of 2nd circle = 10.5 + 10.5 = 21 cm

Radius (r₃) of 3rd circle = 21 + 10.5 = 31.5 cm

Radius (r₄) of 4th circle = 31.5 + 10.5 = 42 cm

Radius (r₅) of 5th circle = 42 + 10.5 = 52.5 cm

Area of gold region = Area of 1st circle =  π*(10.5)*(10.5) cm² = 346.5 cm²

Area of red region = Area of 2nd circle − Area of 1st circle  = π*(21)*(21) – π*(10.5)*(10.5) = 1039.5 cm²

Area of blue region = Area of 3rd circle − Area of 2nd circle =π*(31.5)*(31.5) -π*(21)*(21) = 1732.5 cm²

Area of black region = Area of 4th circle − Area of 3rd circle = π*(42)*(42)-π*(31.5)*(31.5)= 2425.5 cm2

Area of white region = Area of 5th circle − Area of 4th circle=π*(52.5)*(52.5)- π*(42)*(42)= 3118.5 cm²

Hence, areas of gold, red, blue, black, and white regions are 346.5 cm², 1039.5 cm², 1732.5 cm², 2425.5 cm², and 3118.5 cm² respectively.

问题4.汽车的车轮直径均为80厘米。当汽车以每小时66公里的速度行驶时，每个车轮在10分钟内旋转多少圈?

解决方案：

The diameter of the wheel of the car = 80 cm

So the Radius (r) of the wheel of the car = 40 cm

Now Calculating the Circumference of wheel = 2πr = 2π (40) = 80π cm

Speed of car = 66 km/hour

Speed(cm/min)= 110000 cm/min

Distance travelled by the car in 10 minutes = 110000 × 10 = 1100000 cm

Let the number of revolutions of the wheel of the car be p.

So, equating it now 

We get, p × Distance traveled in 1 revolution = Distance traveled in 10 minutes

80π * p = 1100000

p = 4375

Hence, each wheel of the car will make 4375 revolutions.

问题5.勾选以下正确答案，并证明您的选择合理：

如果圆的周长和面积在数值上相等，则圆的半径为

(A)2个单位(B) π单位(C)4个单位(D)7个单位

解决方案：

Let the radius of the circle be r.

Circumference of circle = 2πr

Area of circle = πr²

Also in the question, it is given that, the circumference of the circle and the area of the circle are equal.

Hence, 2πr = πr²

r = 2 units

Hence, the radius of the circle is 2 units.

Option A is the correct answer.