8类RD Sharma解决方案–第3章平方根和平方根–练习3.5

问题1.通过长除法找到以下各项的平方根：

(i)12544

解决方案：

Here, by using long division method,

We found that the square root of 12544 is 112.

(ii)97344

解决方案：

Here, by using long division method 

We found that the square root of 97344 is 312. 

(iii)286225

解决方案：

Here, by using long division method, 

We found that the square root of 286225 is 535.

(iv)390625

解决方案：

Here, by using long division method,

We found that the square root of 390625 is 625. 

(v)363609

解决方案：

Here,  by using long division method,

We found that the square root of 363609 is 603.

(vi)974169

解决方案：

Here, by using long division method,

We found that the square root of 974169 is 987.

(vii)120409

解决方案：

Here, by using long division method,

We found that the square root of 120409 is 347.

(viii)1471369

解决方案：

Here, by using long division method,

We found that the square root of 1471369 is 1213.

(ix)291600

解决方案：

Here, by using long division method,

We found that the square root of 291600 is 540.

(x)9653449

解决方案：

Here, by using long division method,

We found that the square root of 9653449 is 3107.

(xi)1745041

解决方案：

Here, by using long division method,

We found that the square root of 1745041 is 1321.

(xii)4008004

解决方案：

Here, by using long division method,

We found that the square root of 4008004 is 2002.

(十三)20657025

解决方案：

Here, by using long division method,

We found that the square root of 20657025 is 4545.

(十四)152547201

解决方案：

Here, by using long division method,

We found that the square root of 152547201 is 12351.

(十五)20421361

解决方案：

Here, by using long division method,

We found that the square root of 20421361 is 4519.

(xvi)62504836

解决方案：-

Here, by using long division method,

We found that the square root of 62504836 is 7906.

(xvii)82264900

解决方案：

Here, using long division method,

We found that the square root of 82264900 is 9070.

(xviii)3226694416

解决方案：

Here, by using long division method,

We found that the square root of 3226694416 is 56804.

(xix)6475522209

解决方案：

Here, by using long division method,

We found that the square root of 6407522209 is 80047.

(xx)3915380329

解决方案：

Here, by using long division method,

We found that the square root of 3915380329 is 62573.

问题2。找出从以下数字中减去的最小数字，以使其成为一个完美的平方：

(i)2361

解决方案：

Using long division method,

Here, we observed that 2361 is 57 more than the square of 48.

So, we have to subtract 57 from 2361 to make it a perfect square.

(ii)194491

解决方案：

Using long division method,

Here, we observed that 194491 is 10 more than the square of 441.

So, we have to subtract 10 from 194491 to make it a perfect square.

(iii)26535

解决方案：

using long division method,

Here, we observed that 26535 is 291 more than the square of 162.

So, we have to subtract 291 from 26535 to make it a perfect square.

(iv)161605

解决方案：

Using long division method,

Here, we observed that 161605 is 1 more than the square of 402.

So, we have to subtract 1 from 161605 to make it a perfect square.

(v)4401624

解决方案：

Using long division method,

Here, we observed that 4401624 is 20 more than the square of 2098.

So, we have to subtract 20 from 4401624 to make it a perfect square.

问题3.找到必须加到以下数字上的最小数字，以使其成为一个完美的正方形：

(i)5607

解决方案：

Here, by using long division method,

Now, we observe that 5607 is 18 less than square of 75.

Therefore, we must add 18 to 5607 (18 + 5607 = 5625) to get a perfect square of 75 which is 5625. 

(ii)4931

解决方案：

Here, by using long division method,

Now, we observe that 4931 is 110 less than square of 71.

Therefore, we must add 110 to 4931 (110 + 4931 = 5041) to get a perfect square of 71 which is 5041. 

(iii)4515600

解决方案：

Here, by using long division method,

Now, we observe that 4515600 is 25 less than square of 2125.

Therefore, we must add 18 to 4515600 (25 + 4515600 = 4515625) to get a perfect square of 2125 which is 4515625. 

(iv)37460

解决方案：

Here, by using long division method,

Now, we observe that 37460 is 176 less than square of 194.

Therefore, we must add 176 to 37460 (176 + 37460 = 37636) to get a perfect square of 194 which is 37636. 

(v)506900

解决方案：

Here, by using long division method,

Now, we observe that 506900 is 44 less than square of 712.

Therefore, we must add 44 to 506900 (44 + 506900 = 506944) to get a perfect square of 712 which is 506944. 

问题4。找到最大的5位数字，这是一个完美的正方形。

解决方案：

We know that 99999 is the greatest 5-digit number.

Now, to get the greatest 5 digits square number, we have to find the smallest

number that can be subtracted to obtain a perfect square.

So, By using long division method,

Now, the remainder here is 143.

So, 143 must be subtracted from 99999 to get a perfect square.

So, 99999 – 143 = 99856.

Therefore, 99856 is the greatest 5 digits which is a perfect square.

问题5.找到最少的4位数字，这是一个完美的正方形。

解决方案：

We know that 1000 is the smallest 4-digit number.

Now, in order to obtain the least 4 digits perfect square number, we have to find the 

smallest number that can be added to obtain a perfect square. 

So, By using long division method,

Now, we have to add 24 to 1000 to get a perfect square number.

So, 24 + 1000 = 1024

Therefore, 1024 is the least perfect square number with 4 digits which is a square of 32.

问题6.找到最少的六位数，即一个完美的正方形。

解决方案：

We know that 100000 is the smallest 6 digit number.

Now, in order to obtain the least 6 digits perfect square number, we have to find 

the smallest number that can be added to obtain a perfect square.

So, By using long division method,

Now, we have to add 489 to 100000 to get a perfect square number.

So, 489 + 100000 = 100489

Therefore, 100489 is the least perfect square number with 6 digits which is a square of 317.

问题7。找出最大的4位数字，这是一个完美的正方形。

解决方案：

We know that 9999 is the greatest 4 digit number.

Now, to get the greatest 4 digits square number, we have to find the smallest number

that can be subtracted to obtain a perfect square.

So, By using long division method,

Now, the remainder here is 198.

So, 198 must be subtracted from 9999 to get a perfect square.

So, 9999 – 198 = 9801.

Therefore, 9801 is the greatest 4 digits which is a perfect square number.

问题8：将军将士兵们排成一排，形成一个完美的正方形。他发现这样做有60名士兵被排除在外。如果士兵总数为8160，请问每行有多少士兵?

解决方案：

Given, Total number of Soldiers = 8160

Number of left out Soldiers = 60

So, number of soldiers arranged in rows to form a perfect square is 8100 (8160 – 60 = 8100)

So, the number of soldiers in each row = square root of 8100

So, √8100 = 90

Therefore, there are 90 soldiers in each row.

问题9.正方形区域的面积为60025 m ² 。一个人以18公里/小时的速度沿边界骑行。他将在多少时间后返回起点?

解决方案：

Given, area of square field = 60025 m²

Speed of the cyclist = 18 km/h

= 18 × (1000/60 × 60)

= 5 m/s² 

Area of square field = 60025 m²

or, (side)² = 60025

or, side = √60025

or, side = 245 m 

We know that, Total length of boundary = perimeter 

= 4 * side (Perimeter of square = 4 * side of square) 

 = 4 × 245

= 980 m

Hence, Time taken to return to the starting point = 980/5 (time = distance / speed)

= 196 seconds

= 3 minutes 16 seconds

问题10：平整和翻新方形草坪的成本为每m ² 2.50卢比，为1332.50卢比。查找围栏的成本为每米5卢比。

解决方案：

Given, The cost of leveling and turning a square lawn = Rs 2.50 per m²

Total cost of leveling and turning a square lawn = Rs13322.50

Now, Area of square lawn = total cost of leveling and turning a square lawn / the cost of leveling and turning a square lawn

= 13322.50 / 2.50

= 5329 m²

So, the length of side of square = √area of square (As area of square = side²)

= √5329 = 73 m

So, Total length for fencing = Perimeter (As perimeter of square = 4 * side)

= 4 * 73 = 292 m

Therefore, cost of fencing the square lawn = 5 * 292 

= Rs. 1460

问题11：找出最大的三位数是一个完美的正方形。

解决方案：

We know that 999 is the greatest 3-digit number.

Now, to get the greatest 3 digits square number, we have to find the smallest number 

that can be subtracted to obtain a perfect square.

So, By using long division method,

Now, the remainder here is 38.

So, 38 must be subtracted from 999 to get a perfect square.

So, 999 – 38 = 961.

961 = 31²

Therefore, 961 is the greatest 3 digits which is a perfect square number.

问题12.找到必须加到2300的最小数字，以使其成为一个完美的平方。

解决方案：

We will first find the square root of the given number by using the long division method,

Now, we come to know that 2300 is 4 less than square of 48.

So, 48² = 2304

Hence, 4 must be added to 2300 in order to get a perfect square.