Mr. Morris is going to save money and replace his sailboat's mainsail himself. He must determine the area of the mainsail in order to buy the correct amount of material. Calculate the area of the parallelogram to determine how much material should be purchased. Be sure to explain how to decompose this shape into rectangles and triangles. Describe their dimensions and show your work.

Geometric shapes are mathematical shapes that include squares and triangles

The side lengths of a square are congruent, and the adjacent sides are perpendicular.

So, we start by calculating the side lengths using the following distance formula

[tex]d = \sqrt{(x_1 -x_2)^2 + (y_1 -y_2)^2}[/tex]

Using the above formula, we have:

[tex]AB = \sqrt{(3-2)^2 + (4+2)^2} = \sqrt{37[/tex]

[tex]BC = \sqrt{(2+4)^2 + (-2+1)^2} = \sqrt{37[/tex]

[tex]CD = \sqrt{(-4+3)^2 + (-1-5)^2} = \sqrt{37}[/tex]

[tex]DA = \sqrt{(-3-3)^2 + (5-4)^2} = \sqrt{37}[/tex]

The above shows that the side lengths of the square are congruent.

Next, calculate the slope of the sides using:

[tex]m = \frac{y_2 -y_1}{x_2 -x_1}[/tex]

So, we have:

[tex]m_{AB} = \frac{4 +2}{3 -2} = 6[/tex]

[tex]m_{BC} = \frac{-2 + 1}{2+4} = -\frac 16[/tex]

[tex]m_{CD} = \frac{-1 -5}{-4+3} = 6[/tex]

[tex]m_{DA} = \frac{5 -4}{-3-3} = -\frac 16[/tex]

Notice that the opposite slopes are congruent, and the adjacent slopes are opposite reciprocal.

The above highlight, and the equal side lengths show that the figure (1) is a square

The legs of a right isosceles triangle are congruent, and the legs are perpendicular.

So, we start by calculating the lengths of the legs using the following distance formula

[tex]d = \sqrt{(x_1 -x_2)^2 + (y_1 -y_2)^2}[/tex]

Using the above formula, we have:

[tex]XY = \sqrt{(5-4)^2 + (-1-4)^2} = \sqrt{26[/tex]

[tex]XZ = \sqrt{(5-0)^2 + (-1+2)^2} = \sqrt{26[/tex]

The above shows that the legs of the right isosceles triangle are congruent.

Next, calculate the slope of the legs using:

[tex]m = \frac{y_2 -y_1}{x_2 -x_1}[/tex]

So, we have:

[tex]m_{XY} = \frac{5-4}{-1 -4} = -\frac{1}{5}[/tex]

[tex]m_{XZ} = \frac{5-0}{-1+2} = 5[/tex]

Notice that the slopes are opposite reciprocal.

The above highlight, and the equal legs show that the figure (2) is a right isosceles triangle

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Answer:

an equation is said to have no real solutions when value of x < 0

Step-by-step explanation:

here,

(x-4) 2= -9

2x - 8 = -9

2x = -1

x = -1/2

which is less than 0

The equation (x – 4)² = -9 has no real roots.

The solution of a quadratic equation is the value of the variable for which the expression becomes true.

Given,

          (x – 4)² = -9

The LHS of the equation is either positive or zero.

But, the RHS of the equation is a negative value.

Therefore, RHS and LHS can never be equal.

Hence, the equation (x – 4)² = -9 has no real roots.

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Answer:

-1/6

Step-by-step explanation:

5a+2/3=-1/6

5a=-1/6-2/3

5a=-1/6-4/6

5a=-5/6

a=(-5/6)/5

a=(-5/6)(1/5)

a=-5/30

simplify

a=-1/6

Answer:

9x(7-2)=(9x7)-(9x2)

Step-by-step explanation:

You're just multiplying 9 by the numbers inside the parenthesis... I hope this helps you :)

Answer:

X = 10

Step-by-step explanation:

-3=7-x

-7   -7  

-10=-x

10=x

Answer:

  A, B, D, F

Step-by-step explanation:

The work of solving the standard-form equation that is the difference of squares is properly shown except for Line 4. Statement [F] correctly describes the proper solution, which results in the solution set shown in Statement [B].

The first step was necessary to put the equation into standard form (Statement [D]). That form is the difference of squares (Statement [A]).

The statements that correctly describe the work are A, B, D, F.

Answer:

If he worked 5 hours and his rate is $14/hr, then he earned a total of $70.He mailed 300 postcard, so each postcard mailed is $0.2333

Step-by-step explanation:

mark me brainliest!!

Answer:

25 cents per lemon

Step-by-step explanation:

Simply divide

Answer:

See below

Step-by-step explanation:

When solving real world problems relating to unit price, divide the cost by the amount.

2.5/10

The unit price is 0.25.

-hope it helps

Step-by-step explanation:

1st quadrant

hope it helps you

Answer:

b = 5

Step-by-step explanation:

Area of the triangle = b * 12/2 = 6b

Area of the rectangle = 9 * 12 = 108

Total area = 6b + 108  = 138

6b = 138 - 108 = 30

so b = 30/6 = 5

Answer: 10% of 45 is 4.5

So 30% of 45 is 13.5

Step-by-step explanation:

57cm

because you get to subtract

Answer:

34 Euros

Step-by-step explanation:

15% of 40 is 6.

40 minus 6 = 34

[tex][Hello,BrainlyUser][/tex]

Answer:

£34

Step-by-step explanation:

Given:

A pair of jeans cost £40

Discount of 15%

Question:

How much do the jeans cost after the discount

Solve:

£40 x 15% =6

£40 - 6 = 34

Hence, Jeans cost £34 after the discount

[CloudBreeze]

Answer:

11

Step-by-step explanation:

the pattern is

-3 +4 +1 -3 +4 +1

Answer:

Step-by-step explanation:

Vase:

h = 12 in

r = diameter/2 = 6/2 = 3 in

Volume of vase = πr²h

                            = π(3)²*12

Marbles:

r = 3/2 = 1.5 in

[tex]Volume \ of \ marbles = \dfrac{4}{3}\pi r^{3}\\\\\\Volume \ of \ 7 marbles=7*\dfrac{4}{3}\pi (1.5)^{3}[/tex]

Volume of water = Volume of vase - volume of 7 marbles

                            [tex]= \pi (3)^{2}*12 - 7*\dfrac{4}{3}\pi (1.5)^{3}[/tex]

Answer:

second choice

Step-by-step explanation:

the volume of water in the vase

= volume of the vase up to the height 12 inches - volume of 7 marbles

= π(6/2)²(12) - 7(4/3)π(3/2)³

so answer is the second choice

[tex]\mathbb{FINAL\;ANSWER:}[/tex]

[tex]\huge\boxed{\sf{c=4}}[/tex]

[tex]\mathbb{SOLUTION:}[/tex]

Hi! Hope you are having a nice day!

Let's solve the equation for c.

First, subtract 15 and 8:

15-8+c=11

7+c=11

We can actually solve this equation using mental maths.

7+4=11, right? :)

Hope you could comprehend everything.

#CarryOnLearning

Answer:

1/4

Step-by-step explanation:

In this equation, from the beginning we know two things, that there are two people, and that the whole is 1/2

So we just have to divide! Use this equation whenever dealing with word problems like these and you’re stuck.

Amount of object (Pizza Size) divided by Number of Groups (People) = Amount of object in group

So our equation is 1/2 divided by 2

Calculate this and we will get our final answer of

1/4!

Hope this helped and please award brainliest!

Answer:

90+15x=225

Step-by-step explanation:

90 already saved plus 15 an hour equals 225

Step-by-step explanation:

All of these vectors are in DE Moviere Form, so let put them in Vector notation

To this this, multiply the modulus by the exact values of trig functions,

So using this, Vector u

[tex]u= 3j + 3 \sqrt{3} i[/tex]

Vector V

[tex]v = 2 \sqrt{2} j - 2 \sqrt{2} i[/tex]

Vector W

[tex]w = - 6 \sqrt{3}j + 6i[/tex]

So know we can compete the parts.

For Part A:

We first multiply the vector U by -7 then multiply that by v.

[tex] - 7u = - 21j - 21 \sqrt{3} j[/tex]

Know we do cross product

[tex] - 7u(v) = ( - 21j - 21 \sqrt{3} i)(2 \sqrt{2} j \: + 2 \sqrt{2} i[/tex]

First, multiply the j componets.

Then, do the I componets and then add the product together.

[tex] - 42 \sqrt{2} j - 42 \sqrt{6} i[/tex]

So the answer to part A is the vector above.

Part B:

Let first find the dot product of u and w.

First, the cross product of u and w is

[tex] - 18 \sqrt{3} j + 18\sqrt{3} i[/tex]

The magnitudes of each respective vector is

6 and -12. So if we use dot product formula

[tex] \cos( \alpha ) = \frac{xy}{ |x| |y| } [/tex]

So we get

[tex] \cos( \alpha ) = \frac{ - 18 \sqrt{3} + 18 \sqrt{3} }{ - 72} [/tex]

[tex] \cos(a) = \frac{0}{ - 12} [/tex]

[tex] \cos(a) = 0[/tex]

[tex]a = \frac{\pi}{2} [/tex]

Since the angle is pi/2, the vectors are orthogonal.

Answer:

correct

Step-by-step explanation:

Answer:

[tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx = 2 \arctan x - \frac{5}{2(x^2 + 1)} + C[/tex]

General Formulas and Concepts:

Calculus

Differentiation

Derivative Property [Multiplied Constant]:                                                       [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]

Derivative Property [Addition/Subtraction]:                                                     [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]

Derivative Rule [Basic Power Rule]:

Integration

Integration Rule [Reverse Power Rule]:                                                           [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

Integration Property [Multiplied Constant]:                                                     [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

Integration Property [Addition/Subtraction]:                                                   [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Integration Method: U-Substitution

Step-by-step explanation:

Step 1: Define

Identify.

[tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx[/tex]

Step 2: Integrate Pt. 1

Step 3: Integrate Pt. 2

Identify variables for u-substitution for the remaining integral.

Step 4: Integrate Pt. 3

∴ the integration of the given integral   [tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx[/tex]  is equal to  [tex]\displaystyle \bold{2 \arctan x - \frac{5}{2(x^2 + 1)} + C}[/tex].

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---

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration

Answer:

22 inches

Step-by-step explanation:

Diameter = 2*radius = 2(11) = 22 inches

Answer:

A

Step-by-step explanation:

because A is a Equivalent when you times it by 4

Answer:

Use integrals to find the area between the curves.

32

3

Solve for the first variable in one of the equations, then substitute the result into the other equation.

Point Form:

(

0

,

0

)

,

(

4

,

4

)

Equation Form:

x

=

0

,

y

=

0

x

=

4

,

y

=

4

To find the volume of the solid, first define the area of each slice then integrate across the range. The area of each slice is the area of a circle with radius

f

(

x

)

and

A

=

π

r

2

.

V

=

π

∫

4

0

(

f

(

x

)

)

2

−

(

g

(

x

)

)

2

d

x

where

f

(

x

)

=

−

x

2

+

5

x

and

g

(

x

)

=

x

Simplify the integrand.

Tap for more steps...

V

=

x

4

−

10

x

3

+

24

x

2

Split the single integral into multiple integrals.

V

=

π

(

∫

4

0

x

4

d

x

+

∫

4

0

−

10

x

3

d

x

+

∫

4

0

24

x

2

d

x

)

By the Power Rule, the integral of

x

4

with respect to

x

is

1

5

x

5

.

V

=

π

(

1

5

x

5

]

4

0

+

∫

4

0

−

10

x

3

d

x

+

∫

4

0

24

x

2

d

x

)

Combine

1

5

and

x

5

.

V

=

π

(

x

5

5

]

4

0

+

∫

4

0

−

10

x

3

d

x

+

∫

4

0

24

x

2

d

x

)

Since

−

10

is constant with respect to

x

, move

−

10

out of the integral.

V

=

π

(

x

5

5

]

4

0

−

10

∫

4

0

x

3

d

x

+

∫

4

0

24

x

2

d

x

)

By the Power Rule, the integral of

x

3

with respect to

x

is

1

4

x

4

.

V

=

π

(

x

5

5

]

4

0

−

10

(

1

4

x

4

]

4

0

)

+

∫

4

0

24

x

2

d

x

)

Combine

1

4

and

x

4

.

V

=

π

(

x

5

5

]

4

0

−

10

(

x

4

4

]

4

0

)

+

∫

4

0

24

x

2

d

x

)

Since

24

is constant with respect to

x

, move

24

out of the integral.

V

=

π

(

x

5

5

]

4

0

−

10

(

x

4

4

]

4

0

)

+

24

∫

4

0

x

2

d

x

)

By the Power Rule, the integral of

x

2

with respect to

x

is

1

3

x

3

.

V

=

π

(

x

5

5

]

4

0

−

10

(

x

4

4

]

4

0

)

+

24

(

1

3

x

3

]

4

0

)

)

Simplify the answer.

Tap for more steps...

V

=

384

π

5

The result can be shown in multiple forms.

Exact Form:

V

=

384

π

5

Decimal Form:

V

=

241.27431579

…

image of graph

To find the volume of the solid, first define the area of each slice then integrate across the range. The area of each slice is the area of a circle with radius

f

(

x

)

and

A

=

π

r

2

.

V

=

π

∫

4

0

(

f

(

x

)

)

2

−

(

g

(

x

)

)

2

d

x

where

f

(

x

)

=

−

x

2

+

5

x

and

g

(

x

)

=

x

Simplify the integrand.

Tap for more steps...

V

=

x

4

−

10

x

3

+

24

x

2

Split the single integral into multiple integrals.

V

=

π

(

∫

4

0

x

4

d

x

+

∫

4

0

−

10

x

3

d

x

+

∫

4

0

24

x

2

d

x

)

By the Power Rule, the integral of

x

4

with respect to

x

is

1

5

x

5

.

V

=

π

(

1

5

x

5

]

4

0

+

∫

4

0

−

10

x

3

d

x

+

∫

4

0

24

x

2

d

x

)

Combine

1

5

and

x

5

.

V

=

π

(

x

5

5

]

4

0

+

∫

4

0

−

10

x

3

d

x

+

∫

4

0

24

x

2

d

x

)

Since

−

10

is constant with respect to

x

, move

−

10

out of the integral.

V

=

π

(

x

5

5

]

4

0

−

10

∫

4

0

x

3

d

x

+

∫

4

0

24

x

2

d

x

)

By the Power Rule, the integral of

x

3

with respect to

x

is

1

4

x

4

.

V

=

π

(

x

5

5

]

4

0

−

10

(

1

4

x

4

]

4

0

)

+

∫

4

0

24

x

2

d

x

)

Combine

1

4

and

x

4

.

V

=

π

(

x

5

5

]

4

0

−

10

(

x

4

4

]

4

0

)

+

∫

4

0

24

x

2

d

x

)

Since

24

is constant with respect to

x

, move

24

out of the integral.

V

=

π

(

x

5

5

]

4

0

−

10

(

x

4

4

]

4

0

)

+

24

∫

4

0

x

2

d

x

)

By the Power Rule, the integral of

x

2

with respect to

x

is

1

3

x

3

.

V

=

π

(

x

5

5

]

4

0

−

10

(

x

4

4

]

4

0

)

+

24

(

1

3

x

3

]

4

0

)

)

Simplify the answer.

Tap for more steps...

V

=

384

π

5

The result can be shown in multiple forms.

Exact Form:

V

=

384

π

5

Decimal Form:

V

=

241.27431579

…

y

=

3

x

+

4

,

[

2

,

3

]

y

=

3

x

−

2

,

(

0

,

4

)

y

=

√

x

,

[

1

,

4

]

Step-by-step explanation:

Answer:

51

Step-by-step explanation:

ΔABC: AC = 5 - (-1) = 6

height = (-1) - (-4) = 3

Area = 6*3/2 = 9

Rectangle ACDE, length AE = 6 - (-1) = 7

height DE = 5 - (-1) = 6

Area = 7 * 6 = 42

So the area of this polygon = 9 + 42 = 51

Answer:

a = 6

b = 7

Step-by-step explanation:

Since both 36 and 49 are perfect squares, you can find their square roots which are 6 and 7.

To check to see if it is correct, replace the a with 6 and replace the b with 7, then evaluate the exponents.

Answer:

Distributive Property

Step-by-step explanation:

If you multiply first,

6(2 + .03)

6•2 + 6•.03

You must multiply the 6 times BOTH of the terms inside the parenthesis.

This is using the Distributive Property.

Answer:

D

Step-by-step explanation:

2.2 : x = 4.1 : 12.3

2.2/x = 4.1/12.3 = 1/3

so x = 2.2 * 3 = 6.6

D

Answer:

87

Step-by-step explanation:

First method:

- Open Excel and enter 360, 353 and 346 in column A (below each other)

- Select all 3 cells and drag the bottom-right square down to row 40.

Second method:

- Use the equation t(n) = 360-7(n-1)

- Solve for n=40

- t(40) = 360 - 7*39 = 87