D. A triangle with sides x, y, and z has a fixed perimeter 2s = x +y+ 2. It's area is given by Heron's formula: A = V s(s – x)(s – y)(s – 2). Show that among all triangles with fixed perimeter, the one with largest area is equilateral. (Hint: maximize A? instead of A.)
D. A triangle with sides x, y, and z has a fixed perimeter 2s = x +y+ 2. It's area is given by Heron's formula: A = V s(s – x)(s – y)(s – 2). Show that among all triangles with fixed perimeter, the one with largest area is equilateral. (Hint: maximize A? instead of A.)
D. A triangle with sides x, y, and z has a fixed perimeter 2s = x +y+ 2. It's area is given by Heron's formula: A = V s(s – x)(s – y)(s – 2). Show that among all triangles with fixed perimeter, the one with largest area is equilateral. (Hint: maximize A? instead of A.)
These two questions are from multivariable calculus.
Transcribed Image Text:D. A triangle with sides x, y, and z has a fixed perimeter 2s = x + y + 2. It's area is given by
Heron's formula:
A = Vs(s – x)(s – y)(s – 2).
|
Show that among all triangles with fixed perimeter, the one with largest area is equilateral.
(Hint: maximize A? instead of A.)
E. The Cobb-Douglass production function for a particular manufacturer is given by
f(x, y) = 100x³/4,1/4
where x represents the units of labor (at $150 per unit) and y represents the units of capital (at
$250 per unit). The total cost of labor and capital is limited to $50,000. Find the maximum
production level for this manufacturer. (The value of the Lagrange multiplier (A) in this problem
has an interesting interpretation. Economists call it the marginal productivity of money. For
each additional dollar spent on production, A additional units of the product can be produced.)
Study of calculus in one variable to multiple variables. The typical operations involved in multivariate calculus are limits and continuity, partial differentiation, and multiple integration. Major applications are in regression analysis, in finance by quantitative analysis, in engineering and social science to study and build high dimensional systems and exhibit deterministic nature.
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