If p, q, r, s and t represent length, breadth, height surface area and volume of a cuboid respectively, then what is \(\frac {1}{p} + \frac {1}{q}+\frac {1}{r}\) equal to?
Understanding Cuboid Dimensions and Variables
This question involves understanding the basic properties of a cuboid, specifically its dimensions, surface area, and volume. We are given variables representing these properties:
p= length of the cuboidq= breadth of the cuboidr= height of the cuboids= surface area of the cuboidt= volume of the cuboid
Our task is to find the value of the expression \(\frac {1}{p} + \frac {1}{q}+\frac {1}{r}\) in terms of the surface area (s) and volume (t).
Key Formulas for Cuboid Calculations
To solve this, we need the standard formulas for the volume and surface area of a cuboid:
- Volume Formula: The volume (\(V\)) of a cuboid is calculated by multiplying its length, breadth, and height. Using the given variables: \[ t = p \times q \times r \]
- Surface Area Formula: The surface area (\(A\)) of a cuboid is the total area of all its faces. It is calculated as twice the sum of the products of the pairs of dimensions (length-breadth, breadth-height, height-length). Using the given variables: \[ s = 2(pq + qr + rp) \]
Evaluating the Target Expression
Let's focus on the expression we need to evaluate: \(\frac {1}{p} + \frac {1}{q}+\frac {1}{r}\).
- Combining Fractions: The first step is to combine the fractions by finding a common denominator, which is \(pqr\). \[ \frac {1}{p} + \frac {1}{q}+\frac {1}{r} = \frac{1 \times qr}{p \times qr} + \frac{1 \times pr}{q \times pr} + \frac{1 \times pq}{r \times pq} \] \[ = \frac{qr}{pqr} + \frac{pr}{pqr} + \frac{pq}{pqr} \] \[ = \frac{qr + pr + pq}{pqr} \]
- Substituting Volume (t): From the volume formula, we know that \(t = pqr\). We can substitute \(t\) into the denominator of our combined fraction: \[ \frac{qr + pr + pq}{pqr} = \frac{qr + pr + pq}{t} \]
- Relating Numerator to Surface Area (s): Now, let's look at the numerator: \(qr + pr + pq\). Compare this to the surface area formula, \(s = 2(pq + qr + rp)\). We can rearrange the surface area formula to isolate the term \(pq + qr + rp\): \[ pq + qr + rp = \frac{s}{2} \]
- Final Substitution and Simplification: Substitute \(\frac{s}{2}\) for the numerator (\(qr + pr + pq\)) and \(t\) for the denominator (\(pqr\)) in our expression: \[ \frac{qr + pr + pq}{t} = \frac{s/2}{t} \] Simplify the resulting complex fraction: \[ \frac{s/2}{t} = \frac{s}{2 \times t} = \frac{s}{2t} \]
Conclusion on the Expression
By using the standard formulas for the volume and surface area of a cuboid and performing algebraic manipulation, we find that the expression \(\frac {1}{p} + \frac {1}{q}+\frac {1}{r}\) simplifies to \(\frac{s}{2t}\).
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