A rain gutter is to be made of aluminum sheets that are 16 inches wide by turning up the edges 90. See the illustration.
​(a) What depth will provide maximum​ cross-sectional area and hence allow the most water to​ flow?
​(b) What depths will allow at least 30 square inches of water to​ flow?

The future value of the investment if the interest is compounded monthly is $89,354.89.

Given that;

Using the compound interest formula;

A = P( 1 + r/n )^(nt)

We plug in our values.

A = 20000( 1 + 0.05/12 )^(12 × 30)

A = 20000( 1 + 0.0041666666666667)^(360)

A = 20000( 1.0041666666666667)^(360)

A = 20000( 4.46774965 )

A = 89354.89

Therefore, the future value of the investment if the interest is compounded monthly is $89,354.89.

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The boat in discuss is 9.03 miles from lighthouse B.

The distance of the boat from lighthouse B as required can be determined by means of the cosine law in which case we have;

c² = 10² + 6² - (2×6×10Cos 63)

c = 81.52

c = 9.03miles.

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The ways to illustrate the shapes are given.

11.  In each new figure, one vertex is introduced, and each vertex is joined to each other vertex.

12.  In each new figure, 2 rows and 1 column are introduced, and the shading at the corner moves 90 degrees clockwise.

13.  In each new figure, one row and one column are introduced, and exactly half of the rectangles are shaded.

14.  In each new figure, except at the bottom vertex, two perpendicular lines are introduced at the remaining vertices.

15.  In each new figure, one inverted white triangle is inscribed in the orange triangle.

16.  The number of squares increases in the pattern. There are blue circles inscribed in one diagonal square and there are yellow circles inscribed in the remaining squares. Also, the blue circles in the diagonal alternate.

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The height of the tide after eight hours is 16 feet.

From the information given, the function is given as 5cos (π/4)t + 11.

In this case, the time is given as eight hours. This will be:

f(8) = 5 × cos (π/4) × 8 + 11

f(8) = 5 × cos(2π) + 11

= (5 × 0.999) + 11

= 16 feet

The height is 16 feet.

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Answer:

A)

x + y = 80

y = x + 20

B) x = 30 minutes

C) No. If Pam dances for 60 minutes, and dances for 20 minutes longer than she does math, then she does math for 40 minutes. This means she spends 100 minutes total. 100 minutes is not 80 minutes, thus this is not possible.

Step-by-step explanation:

A)

x + y = 80

y = x + 20

B) x = 30 minutes

Replacing y with x + 20 in the first equation ->

x + x + 20 = 80

2x + 20 = 80

2x = 60

x = 30

C) No. If Pam dances for 60 minutes, and dances for 20 minutes longer than she does math, then she does math for 40 minutes. This means she spends 100 minutes total. 100 minutes is not 80 minutes, thus this is not possible.

The value of x using the alternate exterior angle theorem is 82°

An equation is an expression that shows the relationship between two or more numbers and variables.

An angle is formed from the intersection of two or more lines. Types of angles are acute, obtuse and scalene.

From the diagram:

x = 82° (alternate exterior angles)

The value of x using the alternate exterior angle theorem is 82°

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based on the given results, the coefficient of variation can be found to be 45.54.

first, find the mean:
= (30 + 60 + 90 + 45) / 4

= 56.25

find the standard deviation using a spreadsheet:

= 25.61737691

the coefficient of variation is therefore:

= standard deviation / mean x 100

= 25.61737691 / 56.25 x 100

= 45.54

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The volume of the glasses is as follows;

The volume of cylinder glass 1 = πr²h

The volume of cylinder glass 1 = 3.14 × 3² × 6

The volume of cylinder glass 1 = 3.14 × 9 × 6

The volume of cylinder glass 1 = 169.56 cm³ = 170 cm³

The volume of the glass 2 = 2 / 3 π r³ + πr²h

The volume of the glass 2 = 2 / 3 × 3.14 × 3³ + 3.14 × 3² × 3

The volume of the glass 2 = 56.52 + 84.78

The volume of the glass 2 = 141.3 cm³ = 141 cm³

The volume of conical glass 3 = 1 / 3 πr²h

The volume of conical glass 3 = 1 / 3 × 3.14 × 3² × (√7² - 3² = √40)

The volume of conical glass 3 = 9.42 × √40

The volume of conical glass 3 = 59.5773111176

The volume of conical glass 3 = 60 cm³

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The area of the shaded part is 52 sq. inches. The correct option is b. 52 sq. inches

From the question, we are to determine the area of the shaded part

In the given diagram, the shaded part is a rectangle

Area of a rectangle = Length × Width

Length of the shaded part = 18 inches - 4 inches - 1 inch = 13 inches

Width of the shaded part = 6 inches - 2inches = 4 inches

Then,

Area of the shaded part = 13 inches × 4 inches

Area of the shaded part = 52 sq. inches

Hence, the area of the shaded part is 52 sq. inches. The correct option is b. 52 sq. inches

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8. The length of the support is 7. 9 m

9. The length of the conveyer is 12m

3. a = 59m

A = 44°

B = 52°

4. c = 88. 6 mm

b = 49. 1 m

B = 34°

8. We have the angle to be 20 degrees

Opposite side = x

hypotenuse = 23m

Using the sine ratio

sin θ = opposite/ hypotenuse

[tex]sin 20 = \frac{x}{23}[/tex]

Cross multiply

[tex]sin 20[/tex] × [tex]23 = x[/tex]

[tex]x = 0. 3420[/tex] × [tex]23[/tex]

x = 7. 9 m

The length of the support is 7. 9 m

9. The angle of elevation is 37. 3 degrees

Hypotenuse = 19 . 0m

Opposite = x

Using the sine ratio

sin θ = opposite/ hypotenuse

[tex]sin 37. 3 = \frac{x}{19}[/tex]

cross multiply

[tex]x = 0. 6059[/tex] × [tex]19[/tex]

x = 11.5

x = 12 m in 2 significant figures

The length of the conveyer is 12m

3. To determine the sides and angles, we use the sine rule;

[tex]\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}[/tex]

For side a, we use the Pythagorean theorem

[tex]c^2 = a^2 + b^2[/tex]

[tex]85^2 = a^2 + 67^2[/tex]

[tex]a = \sqrt{89^2-67^2}[/tex]

[tex]a = \sqrt{3432}[/tex]

a = 58. 58, a = 59m

To find angle A and B, use the sine rule

[tex]\frac{59}{sin A } = \frac{85}{sin 90}[/tex]

cross multiply

[tex]sin A[/tex] × [tex]85[/tex] = [tex]sin 90[/tex] × [tex]59[/tex]

make sin A subject of formula

[tex]sin A = \frac{59}{85}[/tex]

[tex]sin A = 0. 6941[/tex]

A = [tex]sin^-^1(0. 6941)[/tex]

A = 44°

[tex]\frac{67}{sin B} = \frac{85}{sin 90}[/tex]

cross multiply

[tex]sin B[/tex] × [tex]85[/tex] = [tex]sin 90[/tex] × [tex]67[/tex]

make sin b subject of formula

[tex]sin B = \frac{67}{85}[/tex]

[tex]sin B = 0. 7882[/tex]

B = [tex]sin^-^1( 0. 7882)[/tex]

B = 52°

4. To find the sides, we use the sine rule;

[tex]\frac{74. 0}{sin 56. 6} = \frac{c}{sin 90}[/tex]

Cross multiply

[tex]sin 56. 6[/tex] × [tex]c = sin 90[/tex] × [tex]74[/tex]

make 'c' subject of formula

[tex]c = \frac{74}{0. 8348}[/tex]

c = 88. 6 mm

To find length b, we use the Pythagorean theorem

[tex]c^2 = a^2 + b^2[/tex]

[tex]b^2 = c^2 - a^2[/tex]

[tex]b^2 = 88. 8^2 - 74^2[/tex]

[tex]b = \sqrt{7885. 44 - 5476}\\\\ b = \sqrt{2409. 44}[/tex]

b = 49. 1 m

[tex]\frac{74. 0}{sin 56. 6} = \frac{49. 1}{sin B}[/tex]

cross multiply

[tex]sin B = \frac{40. 99}{74. 0}[/tex]

B = [tex]sin^-^1(0. 5539)[/tex]

B = 34°

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The expression x²-10x-21 is wrong since the product of two negative signs is equal to positive.

Quadratic equations are equations that has a leading degree of 2. The standard quadratic equation is given as;

f(x) = ax^2+bx+c

Given the product of the expression below;

f(x) = (x-3)(x-7)

Expand

f(x) = x(x) - 7x - 3x -3(-7)

f(x) = x^2 - 7x - 3x + 21

f(x) = x^2 - 10x + 21

Hence the expression x²-10x-21 is wrong since the product of two negative signs is equal to positive.

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The area of the enclosed figure is 4 square units.

The area is the amount of area occupied by an object's flat (2-D) surface or shape.

The triangle's area can be described as,

Area = 1/2 x base x height.

To find the area of the enclosed figure, we need to first plot the given points and then determine the shape of the figure.

Plotting the points A, B, and C on a coordinate grid, we get:

The image is given below.

Connecting points A, B, and C, we can see that the figure formed is a right triangle with legs of lengths 4 and 2.

Therefore, the area of the enclosed figure is:

Area = (1/2) x base x height

= (1/2) x 4 x 2

= 4 square units

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The complete question:

Maria plots the following points on a coordinate grid. Then, she connects each plotted point to the previous one in the order shown to form a figure. What is the area, in square units, of the enclosed figure?

Point A (1, 3)

Point B  (5, 7)

Point C(5, 3)

Point A (1, 3)

Answer: The graph of f(x) is horizontally stretched

Using a system of equations, it is found that:

A system of equations is when two or more variables are related, and equations are built to find the values of each variable.

In this problem, the variables are:

Peter pays $50 for 1 orchestra and 4 balcony tickets, hence:

a + 4b = 50 -> a = 50 - 4b.

Justine pays $150 for 3 orchestra and 12 balcony tickets, hence:

3a + 12b = 150.

a + 4b = 50 -> a = 50 - 4b.

The equations are equal, hence we suppose that the costs are equal, that is:

a + 4b = 50 -> 5a = 50 -> a = 10.

Hence:

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Answer: 23 inches by 7 inches

Step-by-step explanation:

161 = 7 x 23 (or 1 x 161 but that doesn't help...)

And to check:

7 x 3 + 2 = 23

Answer:

23inches by 7inches

Step-by-step explanation:

7 ×3+2=23

Answer:

A

Step-by-step explanation:

Draw a diagonal line connecting the centre to a vertex at the bottom. This makes a triangle. It is a regular pentagon so the base of that triangle is 7.6 / 2, which is 3.8. You also know that the angle from the apothem to the base is 90 degrees. You also know that you could split the centre angle into 5 triangles. 360 / 5 = 72 and we have half of it so it is 36 degrees.

Let's see what we have so far:

The base of the triangle is 3.8

The angle at the bottom is 90

The angle at the top is 36

We now have the tan and the opposite and we need the adjacent so rearrange for adjacent.

This gives you adjacent = opposite / tan(angle)

adjacent = 3.8 / tan(36)

adjacent = 5.2m

The first step is to find out how question in total she is able to miss, while still passing.

We know that there are [tex]40\\[/tex] questions on the test and she must get [tex]70[/tex]% of the questions correct.

Let's calculate how many questions that is:

40 x 0.70

= 28.

Subtract

40 - 28

= 12.

She is able to get 12 answers incorrect.

Now we must find out how many questions she got wrong when she got 40% of the first 15 correct.

If she got 40% correct; She got 60% incorrect.

Let's do the math:

Multiply:

15 x 0.40

= 6

Subtract:

15 - 6

= 9

This means that she got 6 questions correct, and 9 questions incorrect.

With the total allotment of 12 incorrect answers allowed, she can only get 3 more questions wrong.

Now we must find the percentage of the remaining 25 questions she can get wrong.

She can only get 3 wrong and she must get 22 correct.

Multiply:

22 x 4

= 88.

This means that she has to get 88% of the remaining 25 questions correct.

The nonreal solutions to the polynomial equation are: (2 + i√11, 2 - i√11).

By critically observing the graph given, we can logically deduce that the real solutions of the polynomial are -1 and 4. This ultimately implies that, the given polynomial expression can be evaluated as follows:

(x + 1)(x - 4) = x² - 3x - 4.

x⁴ - 7x³ + 23x² - 29x - 60 = x² - 3x - 4.

Also, the nonreal solutions can be determined by dividing the polynomial expression by the real solutions as follows:

Nonreal solutions = (x⁴ - 7x³ + 23x² - 29x - 60)/(x² - 3x - 4)

Nonreal solutions = (2 + i√11, 2 - i√11).

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Answer:

11

Step-by-step explanation:

(3x-28) + (3x-30) + x = 33

7x - 58 = 33

7x = 91

x = 13

DE = 3x - 28 = 3*13 - 28 = 11

Answer: The graph of g(x) is narrower than f(x) and is shifted to the left 2 units and down 1 unit.

Step-by-step explanation:

See attached image.

The function f(x) = x² is transformed and became g(x) = 3(x+2)² -1 has the graph of g(x) is wider than f(x) and is shifted to the left 2 units and down 1 unit.

By applying suitable graph transformation we can shift, stretch, shrink, a given graph along x and y axis.

According to the given question given a graph f(x) = x² and g(x) is                        3(x+2)² - 1.

Let us break transformations of g(x) one by one.

g(x) = 3f(x) which implies it's amplitude or vertical stretch is increased by a factor of 3.

Now, g(x) = 3(x+2)² implies the graph is shifted towards left by 2 units as x + 2 = 0 is x = -2.

Now, g(x) = 3(x+2)² - 1 implies that the whole graph has been shifted 1 units down along the y axis.

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Answer:

$42,600

Step-by-step explanation:

284000 x .15

Step-by-step explanation:

solution:

given,

or, 2:5 = 8:x

or, 2/5 = 8/x [ changing into fraction form ]

or, 2x = 40 [ doing cross-multiplication ]

or, x = 20 [ dividing both sides with 2]

Answer:

No Solution

Step-by-step explanation:

1) Simplify -6 - 3 to -3.

-3 = -21

2) Since -3 = -21 is false, there is no solution.

No solution

Answer:

$13.50

Step-by-step explanation:

If I am taking off 40%, that means that the dress now cost 60% of the original cost.  .6 x 20 is 12.

You could also calculate the 20% off and subtract that from the original cost.  .4 x 20 =8.  Now 20 - 8 is also 12.

Now, we have to calculate the tax.  10% of 12 is .1 x 12 or $1.20.  We add the tax to the 12 to get 13.50.

Answer:

18.75 percent

Step-by-step explanation:

150 / 800 x 100 = 18.75

Answer: 5

Step-by-step explanation:

[tex]\sqrt{(-4-0)^2 + (1-(-2))^2}=5[/tex]