The sum of the two bases and the altitude of a trapezoid is 16ft. a) Define the area A of the trapezoid as function of its altitude. b) Find the altitude for which the trapezoid has the largest possible are. (Optimization)
a+b+h=16
Area= h(a+b)/2= h(16-h)/2
dA/dh=0= ... you do it and solve.
dA/dh=(2)(1)(-1)-(h(16-h))(0)/4
dA/dh = -2
is this right?
a) To define the area A of the trapezoid as a function of its altitude, let's suppose the two bases of the trapezoid are denoted by b1 and b2, and the altitude is denoted by h.
The formula to calculate the area of a trapezoid is:
A = (1/2) * (b1 + b2) * h
Since the question states that the sum of the two bases and the altitude is 16ft, we can express this as an equation:
b1 + b2 + h = 16
Now, we can solve this equation for one of the bases in terms of the other base and the altitude. Let's solve it for b1:
b1 = 16 - b2 - h
Substituting this value back into the area formula:
A = (1/2) * (16 - b2 - h + b2) * h
A = (1/2) * (16 - h) * h
A = (8 - 0.5h) * h
A = 8h - 0.5h^2
Therefore, the area A of the trapezoid is a function of its altitude h, given by A = 8h - 0.5h^2.
b) To find the altitude for which the trapezoid has the largest possible area, we need to find the maximum value of the area function A = 8h - 0.5h^2.
To find the maximum, we can take the derivative of the function with respect to h and set it equal to zero:
dA/dh = 8 - h = 0
Solving this equation, we find h = 8.
So, the altitude for which the trapezoid has the largest possible area is 8 ft.
a) To define the area A of a trapezoid as a function of its altitude, we need to use the formula for the area of a trapezoid. The area of a trapezoid is given by the formula:
A = (1/2) * (b1 + b2) * h
where b1 and b2 are the lengths of the two bases of the trapezoid and h is its altitude.
In this case, we are given that the sum of the two bases and the altitude is 16 ft. Let's say the length of one base of the trapezoid is x ft. Then the length of the other base would be (16 - x) ft. Thus, we can rewrite the formula for the area as:
A = (1/2) * (x + (16 - x)) * h
= (1/2) * (16) * h
= 8h
So, the area A of the trapezoid is simply 8 times the altitude h.
b) To find the altitude for which the trapezoid has the largest possible area, we need to optimize the area function. Since we already have the area A expressed in terms of the altitude h, we can treat A as a function of h.
The key idea for optimization is that the maximum or minimum value of a function occurs at either a critical point (where the derivative is equal to zero) or at the endpoints of the given interval.
Since h can range from 0 to 16 (based on the given condition), let's find the critical point(s) by taking the derivative of A with respect to h and setting it equal to zero:
dA/dh = 8
Setting this derivative equal to zero gives us no critical points. So, we only need to consider the endpoints of the interval [0, 16].
To find the maximum value of the area A, we evaluate A at the endpoints:
A(0) = 8 * 0 = 0
A(16) = 8 * 16 = 128
Comparing these two values, we can see that A(16) = 128 is the maximum value of the area A.
Therefore, the altitude for which the trapezoid has the largest possible area is h = 16 ft.