Solution: Prime factorizations: $15 = 3 imes 5$, $20 = 2^2 imes 5$. LCM is $2^2 imes 3 imes 5 = 60$. The synchronized cycle length is $oxed{60}$.Question: A civil engineer designs a parabolic arch with the equation $ y = -ax^2 + bx + c $. If the arch's highest point is at $ (3, 10) $ and passes through $ (0, 4) $, find the value of $ a $.

Sun, 10 May 2026 04:01:04 +0000

Solution: Finding the Coefficient $ a $ in the Parabolic Arch Equation When designing elegant architectural structures like parabolic arches, precision in mathematics ensures both structural integrity and aesthetic beauty. In this case, a civil engineer models a parabolic arch using the quadratic equation:
$$ y = -ax^2 + bx + c $$
The arch reaches its highest point at $ (3, 10) $ and passes through the point $ (0, 4) $. We are asked to find the value of $ a $. --- ### Step 1: Use the vertex form of a parabola
Since the vertex (highest point) of the parabola is at $ (3, 10) $, we rewrite the equation in vertex form:
$$ y = -a(x - 3)^2 + 10 $$
(Note the negative sign in front of $ a $ accounts for the downward-opening parabola.)