\(p\) varies directly as \((x^2 + y^2 + z^2)\). When \(x = 1, y = 2, z = 3\), then \(p = 70\). What is the value of \(p\) when \(x = -1, y = 1\), \(z = 5\)?
Direct Variation Problem Explained
This problem involves the concept of direct variation. When one quantity varies directly as another, it means they are proportional. In this specific question, the quantity '\(p\)' varies directly as the sum of the squares of '\(x\)', '\(y\)', and '\(z\)', which can be written as \((x^2 + y^2 + z^2)\).
Mathematically, we express this relationship as:
\( p \propto (x^2 + y^2 + z^2) \)
To turn this proportionality into an equation, we introduce a constant of variation, often denoted by '\(k\)'. The equation becomes:
\( p = k(x^2 + y^2 + z^2) \)
Our goal is to find the value of '\(p\)' for a new set of '\(x\)', '\(y\)', and '\(z\)' values. To do this, we first need to determine the value of the constant '\(k\)' using the initial conditions provided.
Step-by-Step Solution
Step 1: Finding the Constant of Variation (k)
We are given the first set of values:
- \(x = 1\)
- \(y = 2\)
- \(z = 3\)
- \(p = 70\)
Let's substitute these values into our equation \( p = k(x^2 + y^2 + z^2) \) to find '\(k\)'.
First, calculate the value of \((x^2 + y^2 + z^2)\):
\( x^2 + y^2 + z^2 = (1)^2 + (2)^2 + (3)^2 \)
\( = 1 + 4 + 9 \)
\( = 14 \)
Now, substitute this back into the main equation:
\( 70 = k \times 14 \)
To solve for '\(k\)', we divide both sides by 14:
\( k = \frac{70}{14} \)
\( k = 5 \)
So, the constant of variation '\(k\)' is 5. Our specific variation equation is now:
\( p = 5(x^2 + y^2 + z^2) \)
Step 2: Calculating the New Value of p
Now, we need to find the value of '\(p\)' using the second set of values:
- \(x = -1\)
- \(y = 1\)
- \(z = 5\)
We use the equation we found in Step 1:
\( p = 5(x^2 + y^2 + z^2) \)
First, calculate the new value of \((x^2 + y^2 + z^2)\):
\( x^2 + y^2 + z^2 = (-1)^2 + (1)^2 + (5)^2 \)
\( = 1 + 1 + 25 \)
\( = 27 \)
Finally, substitute this value back into the equation to find '\(p\)':
\( p = 5 \times 27 \)
\( p = 135 \)
Conclusion
Using the principles of direct variation and the calculated constant of variation (\(k=5\)), we found that when \(x = -1, y = 1\), and \(z = 5\), the value of '\(p\)' is 135.
Similar Questions
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1. \(P^2 + Q^2\) varies as R.
2. PQ varies as R.
3. \(P^2 - Q^2\) varies as R.
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- If \(x\) varies directly as \(y\), then which of the following is/are correct ?
I. \((x^2+y^2)\) varies directly as \(y^2\).
II. \(\left(\frac{x^3}{y^2}\right)\) varies directly as \(y\).
Select the answer using the code given below : - Two numbers P and Q are such that P : Q = x : y. If 1 is added to both the numbers, the ratio becomes 7 : 10. If 3 is added to both the numbers, the ratio becomes 3 : 4. What is (x + y) : (x - y) equal to ?
Important Questions from Ratio and proportion
A sum of Rs. 2,500 is distributed among X, Y and Z in the ratio 1/2 : 3/4 : 5/6 . What is the difference between the maximum share and the minimum share?
In a class, there are three groups A, B and C. If one student from group A and two students from group B are shifted to group C, then what happens to the average weight of the students of the class?
How many different sums can be formed with the denominations Rs. 50, Rs. 100, Rs. 200, Rs. 500 and Rs. 2,000 taking at least three denominations at a time?
In a 500 metres race, B starts 45 metres ahead of A, but A wins the race while B is still 35 metres behind. What is the ratio of the speeds of A to B assuming that both start at the same time?
The monthly incomes of Peter and Paul are in the ratio of 4 : 3. Their expenses are in the ratio of 3 : 2. If each saves Rs. 6,000 at the end of the month, their monthly incomes respectively are (in Rs.)
