X is directly proportional to the square of Y. What is the percentage change in X, if Y increase by 20%
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Step-by-step solution
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Write the proportionality relationship.
If X is directly proportional to the square of Y, we can write:
X = kY²
where k is the constant of proportionality.Define the original and new values of Y.
Let the original value of Y be Y₁.
Y₁ = Y
Y increases by 20%, so the new value of Y, Y₂, is:
Y₂ = Y₁ + 0.20Y₁
Y₂ = 1.20Y₁Calculate the original and new values of X.
Original X, X₁:
X₁ = kY₁²New X, X₂:
Substitute Y₂ into the proportionality equation:
X₂ = k(Y₂)²
X₂ = k(1.20Y₁)²
X₂ = k(1.44Y₁²)
X₂ = 1.44(kY₁²)
Since X₁ = kY₁², we can write:
X₂ = 1.44X₁Calculate the percentage change in X.
The formula for percentage change is:
Percentage Change = ((New Value - Original Value) / Original Value) × 100%
Percentage Change in X = ((X₂ - X₁) / X₁) × 100%
Substitute X₂ = 1.44X₁:
Percentage Change in X = ((1.44X₁ - X₁) / X₁) × 100%
Percentage Change in X = (0.44X₁ / X₁) × 100%
Percentage Change in X = 0.44 × 100%
Percentage Change in X = 44%The percentage change in X is an increase of 44%.
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