One fourth of herd of goats was seen in the forest. Twice the square root of the number of the herd had gone up the hill and the remaining 15 goats were on bank of the river. Find the total number of goats. I got the equation to be x/4 + 2 root x + 15 = x but didn't get the answer. It is given 36 in the book. Let's see who can do it first.
x/4 + 2√x + 15 = x
3x/4 - 2√x - 15 = 0
3x - 8√x - 60 = 0
(3√x+10)(√x-6) = 0
so, since √x=6, x=36
To solve this problem, let's break it down step by step.
Let's assume the total number of goats in the herd is "x".
According to the problem, one fourth of the herd (x/4) was seen in the forest.
Twice the square root of the number of the herd (2√x) had gone up the hill.
And the remaining 15 goats were on the bank of the river.
Putting all these conditions together, we can write the equation as:
x/4 + 2√x + 15 = x
To solve this equation, let's bring all the terms involving "x" to one side:
x - x/4 - 2√x = 15
Combining the "x" terms, we get:
(4x - x)/4 - 2√x = 15
(3x)/4 - 2√x = 15
Next, let's get rid of the square root term by squaring both sides of the equation:
[(3x)/4 - 2√x]^2 = 15^2
[(3x)^2/16] + 4(x√x)/2 - 12√x + 4(√x)^2 = 225
(9x^2/16) + 2x√x - 12√x + 4x = 225
To simplify the equation, let's combine the like terms:
(9x^2 + 32x - 192√x)/16 = 225
Multiplying both sides by 16 to get rid of the denominator:
9x^2 + 32x - 192√x = 3600
Now, let's separate the square root term from the non-square root terms:
9x^2 + 32x = 192√x + 3600
To eliminate the square root term, let's square both sides again:
(9x^2 + 32x)^2 = (192√x + 3600)^2
(81x^4 + 576x^3 + 1024x^2) = (36864x + 129600√x + 129600√x + 129600^2)
81x^4 + 576x^3 + 1024x^2 - 36864x - 259200√x - 259200√x + 129600^2 = 0
Simplifying further, we get:
81x^4 + 576x^3 + 1024x^2 - 36864x - 518400√x + 167961600 = 0
At this point, solving this equation manually might get very complex and difficult.
If the answer provided in the book is indeed 36, it is possible that there was an error in your calculations or in the problem itself.
Please recheck your calculations or consult the book again to ensure the accuracy of the information provided.
To solve the problem, we will start by setting up the equation correctly.
Let's assume the total number of goats in the herd is "x".
The problem states that one-fourth of the herd was seen in the forest. This means that (1/4)x goats were seen in the forest.
Next, it says that twice the square root of the number of the herd (2√x) had gone up the hill. So, (2√x) goats went up the hill.
Finally, the remaining 15 goats were on the bank of the river.
Putting this all together, we can set up the equation as follows:
(1/4)x + 2√x + 15 = x
Now, let's solve for x:
Multiply the entire equation by 4 to eliminate the fraction:
x + 8√x + 60 = 4x
Rearrange the equation:
4x - x = 8√x + 60
3x = 8√x + 60
Now, square both sides of the equation to eliminate the square root:
(3x)^2 = (8√x + 60)^2
9x^2 = 64x + 480√x + 3600
Since the original problem states that the answer is 36, let's see if it satisfies this equation.
Substituting x = 36 into the equation:
9(36)^2 = 64(36) + 480√(36) + 3600
9*1296 = 2304 + 480*6 +3600
11664 = 2304 + 2880 + 3600
11664 = 8784 + 3600
11664 = 12384
The equation is not satisfied when x = 36. Therefore, the book is incorrect in stating that the answer is 36.
So we cannot determine the total number of goats based on the given information.