1. **State the problem:** We are given two parabolas based on family birthdates. Given: - Father's birth month = 6 (June) - Father's birth year last two digits = 20 - Mother's birth month = 8 (August) - Mother's birth year last two digits = 22 The parabolas are: $$y_1 = -6x^2 + 20$$ $$y_2 = 8x^2 - 22$$ We need to find if they intersect by solving $y_1 = y_2$. 2. **Find intersections by setting $y_1 = y_2$:** Set: $$-6x^2 + 20 = 8x^2 - 22$$ Bring all terms to one side: $$-6x^2 + 20 - 8x^2 + 22 = 0$$ $$(-6x^2 - 8x^2) + (20 + 22) = 0$$ $$-14x^2 + 42 = 0$$ 3. **Solve for $x^2$:** $$-14x^2 = -42$$ $$x^2 = \frac{-42}{-14} = 3$$ 4. **Find $x$ values:** $$x = \pm \sqrt{3}$$ 5. **Find corresponding $y$ values by substituting into $y_1$ or $y_2$** (using $y_1$): $$y = -6(3) + 20 = -18 + 20 = 2$$ So intersections are at: $$\left(\sqrt{3}, 2\right) \quad \text{and} \quad \left(-\sqrt{3}, 2\right)$$ 6. **Since intersection points exist, no need for birthdates from relatives.** 7. **Find the area between the curves:** The area $A$ between $y_1$ and $y_2$ from $x=-\sqrt{3}$ to $x=\sqrt{3}$ is: $$A = \int_{-\sqrt{3}}^{\sqrt{3}} |y_1 - y_2| \, dx = \int_{-\sqrt{3}}^{\sqrt{3}} |(-6x^2 + 20) - (8x^2 - 22)| \, dx$$ Simplify inside the integral: $$(-6x^2 + 20) - (8x^2 - 22) = -6x^2 + 20 - 8x^2 + 22 = -14x^2 + 42$$ Since $-14x^2 + 42 \geq 0$ on the interval, the absolute value can be removed: $$A = \int_{-\sqrt{3}}^{\sqrt{3}} (-14x^2 + 42) \, dx$$ 8. **Calculate the integral:** $$A = \left[ -14 \frac{x^3}{3} + 42x \right]_{-\sqrt{3}}^{\sqrt{3}} = \left(-\frac{14}{3} x^3 + 42x\right) \Bigg|_{-\sqrt{3}}^{\sqrt{3}}$$ Calculate at upper limit $x=\sqrt{3}$: $$-\frac{14}{3} (\sqrt{3})^3 + 42 (\sqrt{3}) = -\frac{14}{3} (3 \sqrt{3}) + 42 \sqrt{3} = -14 \sqrt{3} + 42 \sqrt{3} = 28 \sqrt{3}$$ Calculate at lower limit $x=-\sqrt{3}$: $$-\frac{14}{3} (-\sqrt{3})^3 + 42 (-\sqrt{3}) = -\frac{14}{3} (-3 \sqrt{3}) - 42 \sqrt{3} = 14 \sqrt{3} - 42 \sqrt{3} = -28 \sqrt{3}$$ 9. **Subtract:** $$A = 28 \sqrt{3} - (-28 \sqrt{3}) = 28 \sqrt{3} + 28 \sqrt{3} = 56 \sqrt{3}$$ **Final answer:** The area between the curves is: $$56 \sqrt{3}$$