Optimization Calculus | Mathematics

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Optimization Calculus

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Optimization Calculus

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A rectangle is bounded by the x-axis and the parabola $y=12-x^2$ . What length and width should the rectangle have so that its area is a maximum?

Given the constraint equation above and the optimization equation $A=2xy$ , choose the DERIVATIVE of the merged (combined) equation.

answer choices

$A'=2xy$

$A'=\ \ 24-6x^2$

$A'=2x\left(12-x^2\right)$

$A'=12-2x$

<math><semantics><mrow><msup><mi>A</mi><mo mathvariant="normal">′</mo></msup><mo>=</mo><mn>2</mn><mi>x</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">A'=2xy</annotation></semantics></math>A′=2xy

alternatives

<math><semantics><mrow><msup><mi>A</mi><mo mathvariant="normal">′</mo></msup><mo>=</mo><mtext> </mtext><mn>24</mn><mo>−</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">A'=\ \ 24-6x^2</annotation></semantics></math>A′= 24−6x2

<math><semantics><mrow><msup><mi>A</mi><mo mathvariant="normal">′</mo></msup><mo>=</mo><mn>2</mn><mi>x</mi><mrow><mo fence="true">(</mo><mn>12</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">A'=2x\left(12-x^2\right)</annotation></semantics></math>A′=2x(12−x2)

<math><semantics><mrow><msup><mi>A</mi><mo mathvariant="normal">′</mo></msup><mo>=</mo><mn>12</mn><mo>−</mo><mn>2</mn><mi>x</mi></mrow><annotation encoding="application/x-tex">A'=12-2x</annotation></semantics></math>A′=12−2x

A closed rectangular shipping box with square base is to be made from 120 square inches of cardboard. What dimensions should the box be for maximum volume?

Choose the constraint and optimization equations that represent the problem.

answer choices

$x^2=120$ and $V=x^3$

$2x+y=120$ and $V=xy$

$x^2y=120$ and $V=2x^2+4xy$

$2x^2+4xy=120$ and $V=x^2y$

<math><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>120</mn></mrow><annotation encoding="application/x-tex">x^2=120</annotation></semantics></math>x2=120 and <math><semantics><mrow><mi>V</mi><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">V=x^3</annotation></semantics></math>V=x3

alternatives

<math><semantics><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>120</mn></mrow><annotation encoding="application/x-tex">2x+y=120</annotation></semantics></math>2x+y=120 and <math><semantics><mrow><mi>V</mi><mo>=</mo><mi>x</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">V=xy</annotation></semantics></math>V=xy

<math><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><mo>=</mo><mn>120</mn></mrow><annotation encoding="application/x-tex">x^2y=120</annotation></semantics></math>x2y=120 and <math><semantics><mrow><mi>V</mi><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">V=2x^2+4xy</annotation></semantics></math>V=2x2+4xy

<math><semantics><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mi>y</mi><mo>=</mo><mn>120</mn></mrow><annotation encoding="application/x-tex">2x^2+4xy=120</annotation></semantics></math>2x2+4xy=120 and <math><semantics><mrow><mi>V</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi></mrow><annotation encoding="application/x-tex">V=x^2y</annotation></semantics></math>V=x2y

A square piece of green origami paper that is 6 inches on a side is being made into a gift box (with no lid) by cutting congruent squares out of each corner, folding up the sides, and taping the edges.

What size squares should you cut out for maximum volume? (do the whole problem)

answer choices

I should cut out squares that are 1/2 in by 1/2 in

I should cut out squares that are 1 in by 1 in

I should cut out squares that are 3 in by 3 in

I should not cut out any squares

Farmer Jo has 32 square feet of land in which to make an enclosure for bunnies, chicks, and penguins. (see picture)

Choose the equation that represents this information.

answer choices

$2x+4y=32$

$2x+2y=32$

$xy=32$

$A=3xy$

alternatives