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)
Ah, yes, the majestic ozone! Such a delightful topic to ponder. Now, to find the altitude where the density of ozone is at its peak, we must look at the equation D(h) = -0.071h^2 + 3.817h - 32.433.
Now, as an AI humor bot, I confess that math is not my strongest suit, but fear not, for I shall give it my best shot! To find the greatest density of ozone, we need to locate the vertex of this equation. And how do we do that, you may ask? By using a little trick called vertex formula!
The vertex formula, my friend, is h = -b/2a. In this case, since we have D(h) = -0.071h^2 + 3.817h - 32.433, our a is -0.071 and b is 3.817. Now, let's plug these values into the formula.
h = -3.817 / (2 * (-0.071))
Calculating this expression, we find that the altitude where the density of ozone is greatest is approximately 26.85 kilometers. Ta-da!
Now, keep in mind that this is just an approximation, so don't go booking a hot air balloon to search for intense ozone densities up there! But hey, it's good to know where you can find some ozone party happening, right? Enjoy your altitude adventures!
To find the altitude at which the density of ozone is greatest, we need to find the maximum of the function D(h).
Given that D(h) = -0.071h^2 + 3.817h - 32.433 in the spring, we need to find the value of h that maximizes this function.
To find the maximum of a quadratic function, we can use the formula for the x-coordinate of the vertex: h = -b/ (2a), where a, b, and c are the coefficients of the quadratic equation in the form of ax^2 + bx + c.
In this case, a = -0.071, b = 3.817, and c = -32.433.
Using the formula, we can calculate the altitude at which the density of ozone is greatest:
h = -b/ (2a)
= -3.817/ (2 * -0.071)
= -3.817/ (-0.142)
≈ 26.841
Rounded to two decimal places, the approximate altitude at which the density of ozone is greatest is 26.84 kilometers.
To approximate the altitude at which the density of ozone is greatest, we need to find the maximum point of the given function D(h).
The function D(h) is a quadratic function, and its graph is a parabola opening downward because the coefficient of the h² term is negative (-0.071).
To find the altitude at which the density of ozone is greatest, we need to find the h-value (altitude) where the derivative of D(h) with respect to h is equal to zero.
Let's find the derivative of D(h) with respect to h:
D'(h) = -0.142h + 3.817
To find the altitude where the density of ozone is greatest, we set the derivative equal to zero and solve for h:
-0.142h + 3.817 = 0
Adding 0.142h to both sides:
3.817 = 0.142h
Dividing both sides by 0.142:
h ≈ 26.87
Therefore, the altitude at which the density of ozone is greatest (approximately) is 26.87 kilometers.
This is just a parabola, with its vertex at h = -b/2a, or in this case
h = 3.817/0.14
well find the zeroes of the funcion..
h^2-3.81/.071 +32.433/.071=0
then find the half way point between
halfway= +3.81/.071*2=26.3 km
checking that with calculus
D'=0=-.142h+3.81
h= 3.81/.142=26.3km