Ozone occurs at all levels of Earth's atmosphere. The density of ozone varies both seasonally and latitudinally. At a given city, the density D(h) of ozone (in 10^−3cm/km) for altitudes h between 20 kilometers and 35 kilometers was determined experimentally. For the D(h) and season, approximate the altitude at which the density of ozone is greatest. (Round your answer to two decimal places.)
D(h) = −0.071h^2 + 3.817h − 32.433 (spring)
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To approximate the altitude at which the density of ozone is greatest, we need to find the maximum value of the given function D(h) = -0.071h^2 + 3.817h - 32.433 for the season "spring".
To find the maximum value of the function, we can use calculus. The maximum value occurs at the vertex of the quadratic function. The formula for the x-coordinate of the vertex of a quadratic function is given by x = -b / (2a), where a and b are the coefficients of the quadratic function.
In this case, the function is D(h) = -0.071h^2 + 3.817h - 32.433, so the coefficients are a = -0.071 and b = 3.817.
To find the altitude at which the density of ozone is greatest, we need to find the value of h that corresponds to the x-coordinate of the vertex. Using the formula x = -b / (2a), we can calculate:
h = -3.817 / (2 * -0.071)
≈ 27.01
Therefore, the altitude at which the density of ozone is greatest, for the given city and season, is approximately 27.01 kilometers.