Question

Weight of a sumo is jointly varies as his height and his age. When height is \(1.2 \mathrm{~m}\) and age is 20 years his weight is \(48 \mathrm{~kg}\). Find the weight of the sumo when his height is \(1.5\) metre and age is 30 years : (a) \(60 \mathrm{~kg}\) (b) \(72 \mathrm{~kg}\) (c) \(90 \mathrm{~kg}\) (d) \(58 \mathrm{~kg}\)

Short answer

Answer: (c) 90 kg

Step-by-step solution

1. Set up the proportion equation

According to the problem, the weight of a sumo wrestler is jointly proportional to his height and age. That means, Weight (W) ∝ Height (H) × Age (A). We can rewrite this as Weight = k × Height × Age, where k is the constant of variation/proportionality.

2. Calculate the constant of variation k using the given values

We are given that height is \(1.2 \mathrm{~m}\) and age is \(20 \mathrm{~years}\), and his weight is \(48 \mathrm{~kg}\). Use these values to calculate k: \(48 = k × 1.2 × 20\) To find k, divide both sides by \((1.2 × 20)\): \(k = \frac{48}{1.2 × 20}\) Calculate k: \(k = 2\)

3. Use k to find the weight of the sumo at the new height and age

Now use the constant of variation k and the new height and age values to find the sumo's weight: Height = \(1.5 \mathrm{~m}\), Age = \(30 \mathrm{~years}\); then the equation becomes: \( W = 2 \times 1.5 \times 30 \) Calculate W: \( W = 90 \)

4. Choose the correct answer from the options

Now compare the calculated weight with the given options: (a) \(60 \mathrm{~kg}\) (b) \(72 \mathrm{~kg}\) (c) \(90 \mathrm{~kg}\) (d) \(58 \mathrm{~kg}\) Our calculated weight is \(90 \mathrm{~kg}\), so the correct answer is (c) \(90 \mathrm{~kg}\).

Key concepts

Proportionality Constant

In problems involving joint variation, the proportionality constant is key.
It links the variables in the equation and allows us to solve for unknowns.
For instance, in our exercise about the sumo's weight, the equation is set up as:

• Weight (W) ∝ Height (H) × Age (A).
• This translates to: defined by the formula: \[ W = k \times H \times A \]

Here, \(k\) is the proportionality constant.
It reveals how changes in height and age affect weight.
To find \(k\), we substitute known values into the equation, then solve for \(k\): \[ k = \frac{48}{1.2 \times 20} = 2 \]Having this constant allows us to predict the weight under different conditions.

Mathematical Modeling

Mathematical modeling is a powerful tool that helps us represent real-world scenarios with mathematical equations.
In this context, we use it to understand how a sumo's weight changes with alterations in height and age.
By expressing this relationship as a joint variation equation, we can capture the essence of the problem:

• \( W = k \times H \times A \)

This mathematical model simplifies an otherwise complex relationship, focusing only on relevant factors.
It helps us predict outcomes by substituting the given variables and constants.
Accurate modeling requires a deep understanding of the correlation between variables to make predictions with confidence.

Problem Solving Strategies

Effective problem solving involves strategic thinking and a structured approach.
To tackle joint variation problems, follow these steps:

• Identify the variables and their relationship. Recognize if it's direct, inverse, or joint variation.
• Set up the equation with the proportionality constant.
• Solve for the constant using provided values.
• Use the constant to find unknowns with new inputs.

In our sumo example, these steps guide us to calculate the proportionality constant and then use it to find the sumo's new weight.
Checking your work with given options ensures accuracy.
Such a methodical approach minimizes errors and enhances problem-solving effectiveness.