A rain gutter is to be constructed of aluminum sheets 12 inches wide. After marking off a length of 4 inches from each edge, this length is bent up at an angle θ. The area A of the opening may be expressed as the function: A(θ) = 16 sin θ ⋅ (cos θ + 1). If θ = 45°, what is the area of the opening?

Answer:

5/48

Step-by-step explanation:

1x5=5

8x6=48

which give u

5/48

Answer:

[tex]\large\boxed{\dfrac{5}{48}}[/tex]

Step-by-step explanation:

[tex]\dfrac{1}{8}\times\dfrac{5}{6}=\dfrac{1\times5}{8\times6}=\dfrac{5}{48}\leftarrow\text{this is the simplest form}[/tex]

Answer:

233.33 %.

Step-by-step explanation:

The increase is 10 - 3 = 7c.

Percentage increase =  100 * 7 / 3.

= 700 / 3

=  233.33 %.

The cost of a first-class postage stamp increased approximately 233.33% from 1917 to 1974.

To calculate the percent increase in the cost of a first-class postage stamp from 1917 to 1974, follow these steps:

Thus, the percent increase in the cost of a first-class postage stamp from 1917 to 1974 is approximately 233.33%.

Answer:

sin(45°)= AC/9

cos(45°)= BC/9

Step-by-step explanation:

This is a right angle triangle:

∠ABC =∠CAB = 45°

Now

AC= CB

AB = 9 units.

We will apply sines:

sine(45°)= AC/AB

We know that AB = 9 units.

So substitute the value of side AB

sin(45°)= AC/9

Now apply cos(45°)

cos(45°)=  BC/AB

Again substitute the value of AB:

cos(45°)= BC/9

Thus the answer is

sin(45°)= AC/9

cos(45°)= BC/9  ....

Answer:

A

E

Step-by-step explanation:

Answer:

D

Step-by-step explanation:

For a sequence to be an arithmetic sequence, it must have a common difference. In other words, it must either go down by the same number or up by the same number.

Let's look at the choices:

A. {-10,5,-2.5,1.25,...}

-10 to 5, that went up by 15. It has to keep going up by 15 to be arithmetic. However 5+15 is not -2.5 so it isn't arithmetic.

B. {100,20,4,0.8,...}

100 to 20, that went down by 80. Since 20-80 is not 4, then this sequence is not arithmetic.

C. {1,4,16,48,...}

1 to 4, that went up by 3. 4+3 is not 16 so this is not arithmetic.

D. {-10,-3.5,3,9.5,...}

-10 to -3.5, that went up 6.5.

-3.5+6.5=3

3+6.5=9.5

This is arithmetic. It keeps going up by 6.5.

Answer:

[tex]\frac{x-2}{x^2-x-2}[/tex]

Step-by-step explanation:

A removable discontinuity is when there is a hole in your graph. This is usually because one X value has been canceled out. Most of the time, it takes factoring to figure out if there is a removable discontinuity when looking at an equation.

First, look at the numerator [tex]x-2[/tex] . This can't be factored any further. However, [tex]x^2-x-2[/tex] can be factored since it is a trinomial (has three terms) .

For the purposes of this example, you may want to think about it as

[tex]1x^2 -1x-2[/tex]

To factor, multiply the the outside coefficients

1 x -2 = -2

Now take the middle coefficient (-1) and ask yourself what two numbers multiply to make -2, but still add to be -1.

-2 x 1 = -2

-2 + 1 = -1

So in factored form, the equation is

[tex]\frac{x-2}{(x-2)(x+1)}[/tex]

Since you have x-2 on both top and bottom, that can be canceled out. x - 2 would be your removable discontinuity in this situation.

A removable discontinuity can occur in a function if there are common factors in both the numerator and denominator that can be canceled out.

A function has a removable discontinuity at a particular point if the function is undefined at that point but can be made continuous by redefining the value at that point. To identify the removable discontinuity, we need to factor both the numerator and denominator of the function. By factoring, we can determine if any common factors exist that can be canceled out, resulting in a removable discontinuity.

Let's consider the given functions:

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The logarithm function [tex]\log_ab[/tex], where [tex]a,b>0 \wedge a\not =1[/tex], is increasing for [tex]a\in(1,\infty)[/tex] and decreasing for [tex]a\in(0,1)[/tex]

[tex]\ln x =\log_ex[/tex] and [tex]e\approx 2.7>1[/tex] therefore [tex]\ln x[/tex] is increasing.

Answer:

x = - [tex]\sqrt{7}[/tex]

Step-by-step explanation:

Radical roots occur in pairs, that is

x = [tex]\sqrt{7}[/tex] is a root then so is x = - [tex]\sqrt{7}[/tex]

Answer:

-√7 = -2.64

Step-by-step explanation:

The polynomial function has roots. The first root is 3 and the second is √7.

When we have a square root that means that we get two roots from the same number but one is negative and the other is positive. For example, if we have:

√x² = ±x

Because we can have:

(-x)² = x², or

(x)²=x².

So a square root always gives us two answers, one negative and the other positive.

Step-by-step explanation:

Write a proportion:

2½ cups / 1 batch = x cups / 3¾ batches

Cross multiply:

x × 1 = 2½ × 3¾

To multiply the fractions, first write them in improper form:

x = (5/2) × (15/4)

x = 75/8

Now write in proper form:

x = 9⅜

Your answer is correct!  Well done!

Answer:

{2, 3, √13}  

Step-by-step explanation:

In a right triangle, the sum of the squares of the two shorter sides equals the square of the third side (Pythagoras).

Let's check each set of sides in turn.

A. {2, 3, √13}

2² + 3² = 4 + 9 = 13

(√13)² = 13

This is a right triangle.

B. {5, 5, 2, √10}

This is a quadrilateral (four sides).

C. {5, 12, 15}

5² + 12² = 25 +144 = 169

15² = 225

This is not a right triangle.

Answer:

2) y-x=2

Add x on both sides:

 y        =x+2

The slope is 1 and the y-intercept is 2.

Step-by-step explanation:

So a linear equation in slope-intercept form is y=mx+b where m is the slope and b is the y-intercept.

Let's put all of these in that form:

1) y+x=2

Subtract x on both sides:

 y     =-x+2

The slope is -1 and the y-intercept is 2.

2) y-x=2

Add x on both sides:

 y        =x+2

The slope is 1 and the y-intercept is 2.

3) y-x+2=0

Add x on both sides:

 y     +2=x

Subtract 2 on both sides:

 y        =x-2

The slope is 1 and the y-intercept is -2.

4) y+x-2=0

Add 2 on both sides:

 y +x      =2

Subtract x on both sides:

 y           =-x+2

The slope is -1 and and the y-intercept is 2.

Answer:

d. (-2, 5)

Step-by-step explanation:

Point T is reflected over the y-axis.

Therefore, the coordinates are (-2, 5).

Answer:

T' is (-2,5)

Step-by-step explanation:

To reflect a point across the y-axis, you will note the following:

1- the sign x-coordinate of the point is flipped while the value remains the same

2- both the sign and the value of the y-coordinate remain the same

We are given that:

point T is (2,5)

Applying the above:

sign of x-coordinate is flipped and value is the same :

x- coordinate of T' is -2

both sign and value of y-coordinate are the same:

y-coordinate of T' is 5

Based on the above:

T' would be (-2,5)

Hope this helps :)

Answer:

C. Marc

Step-by-step explanation:

to get the amount they earn in an hour divide the amount they earn by the amount of hours they worked for.

Maria: 60 divided by 8 = 7.50

Marc: 46.50 divided by 6 = 7.75

Answer:

[tex]y=2x-48[/tex]

Step-by-step explanation:

Let

y -----> the profit earned by the hot dog stand daily

x ----> the number of hot dogs sold

we know that

The linear equation that represent this problem is equal to

[tex]y=2x-48[/tex]

This is the equation of the line into slope intercept form

where

[tex]m=2\frac{\$}{hot\ dog}[/tex] ----> is the slope

[tex]b=-\$48[/tex] ---> is the y-intercept (cost of the day's supply)

The question relates to the linear function concept. In context of the problem, the profit earned by the hot dog stand is represented by the equation y = 2x - 48, where 'y' is the profit, 'x' is the number of hot dogs sold, '2' is the profit per hot dog, and '48' is the fixed daily cost.

The question relates to the concept of a linear function in Mathematics. In this case, the profit (y) made by the hot dog stand depends on the number of hot dogs sold (x). The stand has a fixed cost of $48 for each day's supply, and then makes a profit of $2 for each hot dog sold.

The linear function can be represented by the equation y = mx + b, where 'm' is the slope of the line (representing the rate of profit per hot dog sold, which is $2), 'x' is the number of hot dogs sold, and 'b' is the y-intercept (representing the fixed costs of the stand, which is -$48).

Therefore, the equation representing the profit of the hot dog stand for x number of hot dogs sold is: y = 2x - 48.

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Option B, 1/3

Since pepper topping can be in any size pizza, it remains as a 1/3 probability, since there are three toppings per size.

Answer:

Approximately 92.51.

Not sure what the desired rounding is since it isn't listed.

Step-by-step explanation:

So the first vector G is 40.3 m long in a -35 degree direction.

Lat's find the components of G.

[tex]G_x=40.3\cos(-35)=33.0118[/tex].

[tex]G_y=40.3\sin(-35)=-23.1151[/tex].

The second vector H is 63.3 m long in a 270 degree direction.

[tex]H_x=63.3\cos(270)=0[/tex].

[tex]H_y=63.3\sin(270)=-63.3[/tex].

The resultant vector can be found by adding the corresponding components:

[tex]R_x=G_x+H_x=33.0118+0=33.0118[/tex]

[tex]R_y=G_y+H_y=-23.1151+(-63.3)=-86.4151[/tex]

Now we are asked to find the magnitude of [tex](R_x,R_y)[/tex] which is given by the formula [tex]\sqrt{R_x^2+R_y^2}[/tex].

Since [tex](R_x,R_y)=(33.0118,-86.4151)[/tex] then the magnitude is [tex]\sqrt{(33.0118)^2+(-86.4151)^2}=\sqrt{8557.34844}=92.51[/tex].

The magnitude of the sum of vector G (40.3m, -35°) and vector H (63.3m, 270°) is found by breaking each vector into its components, summing these components, and using the Pythagorean theorem. The magnitude of the sum of these vectors is approximately 92.1 m.

Given that vector G has a magnitude of 40.3 m and is in a -35.0° direction, and vector H has a magnitude of 63.3 m and is in a 270° direction, the sum of these vectors can be determined. This sum is found by breaking each vector into its component forms, adding the components together, and then using the Pythagorean theorem to find the magnitude of the result.

For vector G: Gx = 40.3m * cos(-35) = 33m and Gy = 40.3m * sin(-35) = -23.14m. For vector H: Hx = 0 (as sin(270) equals 0) and Hy = -63.3m (as sin(270) equals -1). The sum vector S = (Gx+Hx, Gy+Hy) = (33m+0 , -23.14m-63.3m) = (33m, -86.44m). Thus, to find the magnitude of the sum of the vectors, we use the Pythagorean theorem: |S| = sqrt((33m)² + (-86.44m)²) = 92.1 m (rounded to 1 decimal place).

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Answer:

40

Step-by-step explanation:

Subtracting a number k from 300 looks like this 300-k.

Adding a number k to 220 looks like this 220+k.

They are saying for some number k that we have 300-k and 220+k is the same value.

That is, 300-k=220+k.

This is the equation we are going to solve for your number.

300-k=220+k

Add k on both sides:

300=220+2k

Subtract 220 on both sides;

80=2k

Divide both sides by 2:

40=k

k=40.

So the number is 40.

Check: 300-40=260 while 220+40=260.

Final answer:

The expression 1(y), when y = -5/8, is calculated by multiplying 1 by -5/8, which simply yields -5/8.

Explanation:

The task is to find the value of the expression 1(y) when y is given as -5/8. The expression 1(y) can be interpreted as simply '1 times y'. Therefore, we need to multiply the number 1 by -5/8 to find the answer.

1(y) = 1 * (-5/8) = -5/8

This is a basic arithmetic operation, specifically multiplication, commonly encountered in mathematics. When we multiply any number by 1, the result is the number itself, which in this case applies to the negative fraction -5/8.

Answer:

D.f(x) = −4x + 105

Step-by-step explanation:

Since the function in linear, we know it has a slope.

We know 2 points

(5,85) and (10,65) are 2 points on the line

m = (y2-y1)/(x2-x1)

   = (65-85)/(10-5)

    =-20/5

    =-4

We know a point and the slope, we can use point slope form to write the equation

y-y1 =m(x-x1)

y-85 = -4(x-5)

Distribute

y-85 = -4x+20

Add 85 to each side

y-85+85 = -4x+20+85

y = -4x+105

Changing this to function form

f(x) =-4x+105

Answer: D or f(x) = -4x + 105

Oil added or consumed during the day equals evening oil level minus morning oil level (g - h).

To calculate how much oil was consumed or added during the day, you can use the formula:

Change in oil = g - h

If the value is positive, it means oil was added.

If the value is negative, it means oil was consumed (leaked or used).

For example, if in the morning your car had 5 liters of oil (h = 5) and in the evening it had 7 liters (g = 7):

Change in oil = 7 - 5 = 2 liters

This means 2 liters of oil were added during the day.

Answer:

B

Step-by-step explanation:

Given the 2 equations

y = 2x² - 5x - 7 → (1)

y = 2x + 2 → (2)

Since both equations express y in terms of x we can equate the right sides, that is

2x² - 5x - 7 = 2x + 2 ( subtract 2x + 2 from both sides )

2x² - 7x - 9 = 0 ← in standard form

Consider the factors of the product of the x² term and the constant term which sum to give the coefficient of the x- term

product = 2 × - 9 = - 18 and sum = - 7

The factors are + 2 and - 9

Use these factors to split the x- term

2x² + 2x - 9x - 9 = 0 ( factor the first/second and third/fourth terms )

2x(x + 1) - 9(x + 1) = 0 ← factor out (x + 1) from each term

(x + 1)(2x - 9) = 0

Equate each factor to zero and solve for x

x + 1 = 0 ⇒ x = - 1

2x - 9 = 0 ⇒ 2x = 9 ⇒ x = 4.5

Substitute these values into (2) for corresponding values of y

x = - 1 : y = (2 × - 1) + 2 = - 2 + 2 = 0 ⇒ (- 1, 0)

x = 4.5 : y = (2 × 4.5) + 2 = 9 + 2 = 11 ⇒ (4.5, 11)

Solutions are (4.5, 11) and (- 1, 0)

Answer:

(-1, 0) , (4.5,  11). (the second choice).

Step-by-step explanation:

y = 2x^2 - 5x - 7

y = 2x + 2

Since both right side expressions are equal to y we can equate them.

2x^2 - 5x - 7 = 2x + 2

2x^2 - 7x - 9 = 0

(2x - 9)(x + 1)

x = 4.5 , -1.

Substituting these values of x in the second equation:

When x = -1 , y =2(-1) + 2 = 0.

When x = 4.5, y =  2(4.5) + 2 = 11.

Answer:

You can use teh associative property to split 48 and 82 each into 2 peices

(40+8)(80+2) then you can move the parenthesis around. 40+(8+80)+2

40+(88+2)

40+90=130

Answer:

x(1 - .4)

Step-by-step explanation:

x = regular price.

1 - .4 = .6 = 60%

The sale price is equal to the full price (aka x) minus the discounted price (40% of x = 40/100 times x =  .4x)

Therefore sale price = x - .4x or x(1 - .4)

[tex]\bf \textit{Logarithm Cancellation Rules} \\\\ log_a a^x = x\qquad \qquad a^{log_a x}=x \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \log_6(\sqrt[3]{6})\implies \log_6(6^{\frac{1}{3}})\implies \cfrac{1}{3} \\\\[-0.35em] ~\dotfill\\\\ \log_2(64)\implies \log_2(2^6)\implies 6 \\\\[-0.35em] ~\dotfill\\\\ -3\log_5(25)\implies -3\log_5(5^2)\implies -3(2)\implies -6 \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf \log_2(\sqrt[4]{8})\implies \log_2(\sqrt[4]{2^3})\implies \log_2(2^{\frac{3}{4}})\implies \cfrac{3}{4} \\\\[-0.35em] ~\dotfill\\\\ \log_3\left( \frac{1}{81} \right)\implies \log_3\left( \frac{1}{3^4} \right)\implies \log_3(3^{-4})\implies -4[/tex]

Answer:

3/4 goes with [tex]\log_2(8^\frac{1}{4})[/tex]

-4 goes with [tex]\log_3(\frac{1}{81})[/tex]

-6 goes with [tex]-3\log_5(25)[/tex]

1/3 goes with [tex]\log_6(6^\frac{1}{3})[/tex]

Step-by-step explanation:

[tex]\log_6(6^\frac{1}{3})=\frac{1}{3}\log_6(6)=\frac{1}{3}\cdot 1=\frac{1}{3}[/tex]

[tex]\log_2(64)=6 \text{ since } 2^6=64[/tex]

[tex]-3\log_5(25)=-3(2)=-6 \text{ since } 5^2=25[/tex]

[tex]\log_2(8^\frac{1}{4})=\frac{1}{4}\log_2(8)=\frac{1}{4}\log_2(2^3)=\frac{1}{4}\cdot (3)\log_2(2)=\frac{1}{4} \cdot 3 \cdot 1=\frac{3}{4} [/tex]

[tex] \log_3(\frac{1}{81})=\log_3(\frac{1}{3^4})=\log_3(3^{-4})=-4\log_3(3)=-4(1)=-4[/tex]

Here are few rules I used:

[tex]\log_a(b)=x \text{ means } a^x=b[/tex]

[tex]\log_a(a)=1 [/tex]

[tex]\log_a(b^r)=r \log_a(b)[/tex]

Answer:

x=9

Step-by-step explanation:

Since y=18

The new equation is [tex]2x+3*6=36[/tex]

Moving some terms, it stays like the following:

[tex]2x=36-18[/tex]

solving for x, give us that x=9

Answer:

b

Step-by-step explanation:

Given

3x² - 18x = 21 ( divide all terms by 3 )

x² - 6x = 7

To complete the square

add ( half the coefficient of the x- term )² to both sides

x² + 2(- 3)x + 9 = 7 + 9

(x - 3)² = 16 ( take the square root of both sides )

x - 3 = ± [tex]\sqrt{16}[/tex] = ± 4 ( add 3 to both sides )

x = 3 ± 4, hence

x = 3 - 4 = - 1 and x = 3 + 4 = 7

The correct option is b. [tex]\((x-3)^2=16\); -1 and 7.[/tex]

To solve the given quadratic equation [tex]\(3x^2 - 18x = 21\),[/tex] we first divide the entire equation by 3 to simplify it:

[tex]\[ x^2 - 6x = 7 \]\[ x^2 - 6x + 9 - 9 = 7 \] \[ (x - 3)^2 - 9 = 7 \][/tex]

Now, we isolate the perfect square on one side:

[tex]\[ (x - 3)^2 = 7 + 9 \] \[ (x - 3)^2 = 16 \][/tex]

This is the perfect square equation. To find the solutions, we take the square root of both sides:

[tex]\[ x - 3 = \pm4 \][/tex]

Now, we solve for \(x\) by adding 3 to both sides:

[tex]\[ x = 3 \pm 4 \][/tex]

This gives us two solutions:

[tex]\[ x = 3 + 4 = 7 \] \[ x = 3 - 4 = -1 \][/tex]

 Therefore, the solutions to the equation are [tex]\(x = -1\) and \(x = 7\),[/tex] which corresponds to option b. [tex]\((x-3)^2=16\); -1 and 7.[/tex]

Answer:

C) f(x) = x2

Step-by-step explanation:

Final answer:

To simplify the expression 43 + 2(3 - 2), first solve inside the parentheses (3 - 2 = 1), then multiply by 2 (2 × 1 = 2), and finally add to 43 to get 45.

Explanation:

The question asks to simplify the expression 43 + 2(3 − 2). To simplify, follow the order of operations, often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right). First, solve the expression within the parentheses, which is 3 − 2 = 1. Next, multiply the result by 2 (2 × 1 = 2). Lastly, add this result to 43 to get the final answer, 43 + 2 = 45. This process demonstrates how to approach and simplify mathematical expressions using proper order of operations.

Answer:

[tex]\sqrt{37}[/tex]

Step-by-step explanation:

Distance formula

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

[tex]d = \sqrt {\left( {1 - 2 } \right)^2 + \left( {-2 - 4 } \right)^2 }[/tex]

Simplify

[tex]d = \sqrt {\left( {-1 } \right)^2 + \left( {-6 } \right)^2 }[/tex]

Simplify

[tex]d = \sqrt {\left 1 + \left 36}[/tex]

[tex]d = \sqrt{37}[/tex]

Answer

[tex]d = \sqrt{37}[/tex]

The answer is:

It will take 42.35 minutes to weed the garden together.

To solve the problem, we need to use the given information about the rate for both Laura and her husband. We know that she can weed the garden in 1 hour and 20 minutes (80 minutes) and her husband can weed it in 1 hour and 30 minutes (90 minutes), so we need to combine both's work and calculate how much time it will take to weed the garden together.

So, calculating we have:

Laura's rate:

[tex]\frac{1garden}{80minutes}[/tex]

Husband's rate:

[tex]\frac{1garden}{90minutes}[/tex]

Now, writing the equation we have:

[tex]Laura'sRate+Husband'sRate=CombinedRate[/tex]

[tex]\frac{1}{80}+\frac{1}{90}=\frac{1}{time}[/tex]

[tex]\frac{1*90+1*80}{7200}=\frac{1}{time}[/tex]

[tex]\frac{170}{7200}=\frac{1}{time}[/tex]

[tex]\frac{17}{720}=\frac{11}{time}[/tex]

[tex]\frac{17}{720}=\frac{1}{time}[/tex]

[tex]\frac{17}{720}*time=1[/tex]

[tex]time=1*\frac{720}{17}=42.35[/tex]

Hence, we have that it will take 42.35 minutes to weed the garden working together.

Have a nice day!

Answer:

a. a correlation matrix

Step-by-step explanation:

The best fitting straight line for a data set of X-values plotted against y-values is  called a correlation matrix.