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zloy xaker [14]
3 years ago
10

Let u = 〈1,5〉 and v = 〈3, −2〉. Write each vector in component form and draw an arrow to represent the vector.

Mathematics
1 answer:
Alexxx [7]3 years ago
7 0

Answer:

a. 〈4,3〉

b. 〈-2,7〉

c. 〈11,4〉

Step-by-step explanation:

a. u + v

〈1,5〉 + 〈3, −2〉

〈1+3,5-2〉

〈4,3〉

b. u - v

〈1,5〉 - 〈3, -2〉

〈1-3,5-(-2)〉

〈-2,7〉

c. 2u + 3v

2*〈1,5〉 + 3*〈3,-2〉

〈2*1,2*5〉 + 〈3*3,3*(-2)〉

〈2,10〉 + 〈9,-6〉

〈2+9,10-6〉

〈11,4〉

To draw the vectors, let's put them on the origin (0,0), so let's draw the line until the point of its component form, with an arrow. The vectors are shown below.

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Answer:

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Step-by-step explanation:

Refer to the attached diagram:

The volume of the box is given by

V = Length \times Width \times Height \\\\

Let x denote the length of the sides of the square as shown in the diagram.

The width of the shaded region is given by

Width = 3 - 2x \\\\

The length of the shaded region is given by

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So, the volume of the box becomes,

V =  \frac{1}{2} (5 - 3x) \times (3 - 2x) \times x \\\\V =  \frac{1}{2} (5 - 3x) \times (3x - 2x^2) \\\\V =  \frac{1}{2} (15x -10x^2 -9 x^2 + 6 x^3) \\\\V =  \frac{1}{2} (6x^3 -19x^2 + 15x) \\\\

In order to maximize the volume enclosed by the box, take the derivative of volume and set it to zero.

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We are left with a quadratic equation.

We may solve the quadratic equation using quadratic formula.

The quadratic formula is given by

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Where

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Volume of the box at x= 0.53:

V =  \frac{1}{2} (5 – 3(0.53)) \times (3 - 2(0.53)) \times (0.53) \\\\V = 1.75 \: cm^3

The volume of the box is maximized when x = 0.53 cm

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