A rocket's altitude in meters after t seconds is modeled by the function \( h(t) = -5t^2 + 150t + 100 \). What is the maximum altitude the rocket reaches?

["Title: Understanding Rocket Altitude: How to Find Maximum Height Using an Altitude Model", "Meta Description:\nExplore how the altitude of a rocket modeled by the quadratic function ( h(t) = -5t^2 + 150t + 100 ) reaches its peak. Discover the maximum height and the time it occurs.", "---", "### Reaching New Heights: Maximizing Rocket Altitude Using Algebra", "When launching a rocket, one of the most critical questions is: how high will it go? Engineers rely on mathematical models to predict flight paths—and in the case of the altitude function for a rocket given by\n[\nh(t) = -5t^2 + 150t + 100,\n]\nthe path is a downward-opening parabola. This shape tells us the rocket climbs to a maximum altitude before descending.", "To find that peak altitude, we apply a key principle from algebra: the vertex of a parabola. For any quadratic function in the form\n[\nh(t) = at^2 + bt + c,\n]\nthe time ( t ) at which the maximum (or minimum) height occurs is given by\n[\nt_{\ ext{max}} = -\frac{b}{2a}.\n]", "In our function, ( a = -5 ), ( b = 150 ), and ( c = 100 ). Plugging in the values:\n[\nt_{\ ext{max}} = -\frac{150}{2 \ imes (-5)} = -\frac{150}{-10} = 15 \ ext{ seconds}.\n]\nSo, the rocket reaches its highest point 15 seconds after launch.", "To find the maximum altitude, substitute ( t = 15 ) back into the original function:\n[\nh(15) = -5(15)^2 + 150(15) + 100.\n]\nCalculate step-by-step:\n- ( 15^2 = 225 )\n- ( -5 \ imes 225 = -1125 )\n- ( 150 \ imes 15 = 2250 )\n- Add all terms:\n[\nh(15) = -1125 + 2250 + 100 = 1225 \ ext{ meters}.\n]", "### Conclusion\nThe rocket reaches a maximum altitude of 1225 meters at 15 seconds after launch. Understanding this maximum height helps optimize rocket designs, improve safety, and plan recovery missions. This quadratic model exemplifies how math brings science and engineering to new heights.", "Keywords: rocket altitude model, maximum height formula, parabola rocket trajectory, quadratic function rocket, how high does a rocket go, aerospace engineering math, altitude vs time graph, vertex of a parabola, aerospace altitude predicted", "---", "This concise SEO article explains the concept clearly, uses targeted keywords for search visibility, and guides readers from the formula to the final clear answer."]