Question

A researcher estimates the relationship between reproductive success (𝑁, number of offspring) and horn length (𝐻, in cm) in a wild goat as 𝑁 = 40 − 2.2𝐻 + 0.04𝐻 2 Horn length typically varies from 10 cm to 50 cm in this species. Which one of the following graphs correctly represents this relationship?

• A. P
• B. Q
• C. R
• D. S

Hint:

Expected heterozygosity under HWE is \(H=1-\sum p_i^2\). With a fixed number of alleles, it’s maximized when all alleles are \emph{equally frequent}.

Answer 1

The Correct Option is B

Goat Horn Length vs Reproductive Success

Step 1: Identify functional form and concavity.

\[ N(H) = aH^2 + bH + c, \quad a = 0.04 > 0, \; b = -2.2, \; c = 40 \] Since \(a > 0\), the parabola opens upwards (convex). The curve is U-shaped with a single minimum.

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Step 2: Locate the vertex (minimum).

Derivative method: \(\frac{dN}{dH}\) =\( -2.2 + 0.08H\) \(;\Rightarrow\; H\)= \(\frac{2.2}{0.08}\) = 27.5 \(\text{cm}\)  Completing the square: \[ N(H) = 0.04(H-27.5)^2 + 9.75 \] Thus, \(N_{\min} = 9.75\) at \(H=27.5\) cm.

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Step 3: Behavior in biological range \(10 \leq H \leq 50\).

Since \(10 < 27.5 < 50\), the curve decreases for \(10 \leq H < 27.5\) and increases for \(27.5 < H \leq 50\).

Endpoint values: \[ N(10) = 22, \quad N(50) = 30 \] So, the pattern is clearly U-shaped.

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Step 4: Match with given sketches.

• P: strictly increasing line → not quadratic
• R: strictly decreasing line → not quadratic
• S: inverted parabola (\(\cap\)) → concave down, not ours
• Q: U-shaped parabola (convex) with minimum inside range → correct

Final Answer: \[ \boxed{(B)\; Q} \]