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Question:

Let S(α) = {(x,y): y² ≤ x, 0 ≤ x ≤ α} and A(α) is the area of the region S(α). If for a λ, 0 < λ < 4, A(λ):A(4) = 2:5, then λ equals?

4(425)13

2(425)13

2(25)13

4(25)13

Solution:

S(α) = {(x,y): y² ≤ x, 0 ≤ x ≤ α}
A(α) = 2∫₀ˣ√x dx = 2x³/²
A(4) = 2 × 4³/² = 16
A(λ) = 2 × λ³/²
A(λ)/A(4) = 2/5
=> 2λ³/²/16 = 2/5
=> λ³/² = 16/5
=> λ = (16/5)^(2/3) = (4²/5)^(2/3) = 4^(2/3) 5^(-2/3) = 4(4/25)^(1/3) = 4(425)^(-1/3) = 4(4/25)^(1/3) = 4(25/4)^(1/3) = 2(25/4)^(-1/3) = 2(4/25)^(1/3) = 2(425)^(1/3)