The height of a rectangular garden, modeled by the polynomial \( h(x) = -0.5x^{2} + 6x + 10 \), represents its growth over time. Calculate when the height of the garden will reach its maximum value.

To find the maximum height of the garden, we need to determine the vertex of the quadratic polynomial \( h(x) = -0.5x^{2} + 6x + 10 \). The \( x \)-coordinate of the vertex for a quadratic \( ax^{2} + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = -0.5 \) and \( b = 6 \). Calculating gives us \( x = -\frac{6}{2 \times -0.5} = 6 \). This means that the height reaches its maximum value at \( x = 6 \) (which could represent, say, 6 months or 6 years after planting, depending on the context!). To find the maximum height, substitute \( x = 6 \) back into the polynomial: \( h(6) = -0.5(6^{2}) + 6(6) + 10 = -0.5(36) + 36 + 10 = -18 + 36 + 10 = 28 \). So, the maximum height of the garden is 28 units (maybe centimeters or inches!). Not only does this polynomial capture the beauty of growth, but it also relates closely to parabolic trajectories seen in real-world scenarios, such as projectile motion. Think of when you toss a ball or shoot a basketball; just like the height of the garden, the ball’s height can be modeled using a quadratic equation! Gardening tips often suggest keeping a close eye on seasonal changes, just like monitoring the function's behavior. Remember, over-fertilization can lead to rapid growth at first, but too much can eventually cause your plants to "go to seed" earlier than desired, much like how a polynomial may start very high but then can drop off rapidly! So, keep an eye on your garden—as our function shows, growth can be a beautiful journey, but if not managed right, it could lead to unexpected declines!