Proportions in Triangles (5)

Answer:

The price per pound of bananas is $0.5 and the price per pound of apples is $0.6

Step-by-step explanation:

Let

x -----> the price per pound of bananas

y -----> the price per pound of apples

we know that

12x+10y=12 -----> equation A

8x+5y=7 ----> equation B

Solve the system of equations by graphing

Remember that the solution is the intersection point both graphs

The intersection point is (0.5,0.6)

see the attached figure

therefore

The price per pound of bananas is $0.5

The price per pound of apples is $0.6

To find the price per pound of bananas and apples, we set up and solved a system of equations based on two purchases. We found that bananas are $0.50 per pound and apples are $0.60 per pound.

To determine the price per pound of bananas and apples, we need to set up a system of equations based on the information given. Henrietta's purchase can be represented by the equation 12b + 10a = 12, where b is the cost of bananas per pound and a is the cost of apples per pound. Gustavo's purchase can be represented by the equation 8b + 5a = 7.

Now, let's solve the system of equations:

Multiplying Equation 2 by 2 gives us 16b + 10a = 14, which can be compared to Equation 1 to eliminate the apple's cost:

Subtracting Equation 1 from the doubled Equation 2:

16b - 12b + 10a - 10a = 14 - 12

4b = 2

b = 0.50

Now that we have the cost of bananas per pound, we can substitute b = 0.50 into either Equation 1 or 2 to find the cost of apples per pound. Using Equation 2:

8(0.50) + 5a = 7

4 + 5a = 7

5a = 3

a = 0.60

The price per pound of bananas is $0.50, and the price per pound of apples is $0.60.

Answer:

Step-by-step explanation:

We can prove it through different facts but we will use the fact of  alternate interior angles formed by a transversal with two parallel lines are congruent.

Look at the figure for brief understanding.

Construct a line through B parallel to AC. Angle DBA is equal to CAB because they are a pair of alternate interior angle(alternate interior angles are two interior angles which lie on different parallel lines and on opposite sides of a transversal) The same reasoning goes with the alternate interior angles EBC and ACB....

The sum of the interior angles of a triangle is 180 degrees, which can be demonstrated using properties of parallel lines, alternate interior angles, and Euclidean geometry axioms. By drawing a parallel line and using congruent angles, we show that the sum of angles in a triangle aligns with the angle sum on a straight line.

One classic proof that the sum of the interior angles of a triangle is 180 degrees involves drawing a parallel line to one side of the triangle through the opposite vertex. Let's name the vertices of our triangle A, B, and C. Extend a line from vertex C that is parallel to the line AB. This forms alternate interior angles with the angles at vertices A and B, which we know are equal because of the properties of parallel lines.

Call the angles at A and B in the triangle, angle A and angle B, respectively. Outside of the triangle, we have angles formed between the extended line and the lines AC and BC, let's call these angles A' and B'. By the property of parallel lines and angles, angle A is congruent to angle A' and angle B to angle B'. The line through C forms a straight line, so angle A' plus angle C plus angle B' must equal 180 degrees. Since angle A is congruent to angle A' and angle B to angle B', we can then say that angle A plus angle B plus angle C equals 180 degrees. This is because the sum of the interior angles at A and B and the newly defined angle C is equal to the sum along the straight line, which is always 180 degrees.

Moreover, considering the Euclidean geometry axioms, we know that the sum of angles in a triangle is inherently 180 degrees, and this can be seen in the equilateral triangle example where if we take the large triangle and divide it into four smaller congruent triangles, each of these smaller triangles also has the property that the sum of its angles equals 180 degrees. When we add up the angles from the four small triangles and subtract the sum of the straight angles formed at the large triangle's sides, the result confirms the sum for the large triangle is equivalent to four times the sum for one small one, reinforcing the 180-degree sum rule for each triangle.

Answer:

11≤√134≤12

Step-by-step explanation:

11^2 is 121

and 12^2 is 144

so √134 would have to fall between these numbers

The only thing you can do with this expression is to factor a 5 out of the two terms: we have

[tex]15n-20 = 5(3n-4)[/tex]

Answer:

5(3n-4)

Step-by-step explanation:

because(5*3n)-(5*4)=15n-20

Answer:True

Step-by-step explanation:

Given Fran earn [tex]\$ 225[/tex] per week working 15 hr

i.e. in 7 days he earn [tex]\$ 225[/tex]

in 1 day [tex]\frac{225}{7}[/tex]

i.e. in 15 hr he earns [tex]\frac{225}{7}[/tex]

in 1 hr  [tex]\$ \frac{15}{7}[/tex]

he has to earn [tex]\$600 [/tex]extra to make [tex]\$1400[/tex]

i.e. he needs to work [tex]\frac{600\times 7}{15}[/tex]hr extra

For 20 weeks he needs to work 2 hr extra

i.e. total 17 hr per day to save [tex]\$ 1400[/tex]

she needs to work atleast 17 hr

Answer:

B on edgenuity

Step-by-step explanation:

Answer:

det(A) = (-6)(-2) - (-4)(-7)

Step-by-step explanation:

The determinat of the following matrix:

[tex]\left[\begin{array}{ccc}a&b\\c&d\\\end{array}\right][/tex]

Is given by: Determinant a*d - b*c

In this case, a=-6, b=-7, c=-4 and d=-2.

Therefore the determinant is: (-6)(-2) - (-7)(-4).

Therefore, the correct option is the third one:

det(A) = (-6)(-2) - (-4)(-7)

Answer:

C det(A) = (–6)(–2) – (–4)(–7)

Step-by-step explanation:

EDGE 2020

~theLocoCoco

Answer:

1. reflection across x-axis

2. translation 6 units to the right and 3 units up (x+6,y+3)

Step-by-step explanation:

The trapezoid ABCD has it vertices at points A(-5,2), B(-3,4), C(-2,4) and D(-1,2).

First transformation is the reflection across the x-axis with the rule

(x,y)→(x,-y)

so,

Second transformation is translation 6 units to the right and 3 units up with the rule

(x,y)→(x+6,y+3)

so,

Answer:

Step-by-step explanation:

In all of these problems, the key is to remember that you can undo a trig function by taking the inverse of that function.  Watch and see.

a.  [tex]sin2\theta =-\frac{\sqrt{3} }{2}[/tex]

Take the inverse sin of both sides.  When you do that, you are left with just 2theta on the left.  That's why you do this.

[tex]sin^{-1}(sin2\theta)=sin^{-1}(-\frac{\sqrt{3} }{2} )[/tex]

This simplifies to

[tex]2\theta=sin^{-1}(-\frac{\sqrt{3} }{2} )[/tex]

We look to the unit circle to see which values of theta give us a sin of -square root of 3 over 2.  Those are:

[tex]2\theta =\frac{5\pi }{6}[/tex] and

[tex]2\theta=\frac{7\pi }{6}[/tex]

Divide both sides by 2 in both of those equations to get that values of theta are:

[tex]\theta=\frac{5\pi }{12},\frac{7\pi }{12}[/tex]

b.  [tex]tan(7a)=1[/tex]

Take the inverse tangent of both sides:

[tex]tan^{-1}(tan(7a))=tan^{-1}(1)[/tex]

Taking the inverse tangent of the tangent on the left leaves us with just 7a.  This simplifies to

[tex]7a=tan^{-1}(1)[/tex]

We look to the unit circle to find which values of a give us a tangent of 1.  They are:

[tex]7\alpha =\frac{5\pi }{4},7\alpha =\frac{\pi }{4}[/tex]

Dibide each of those equations by 7 to find that the values of alpha are:

[tex]\alpha =\frac{5\pi}{28},\frac{\pi}{28}[/tex]

c.  [tex]cos(3\beta)=\frac{1}{2}[/tex]

Take the inverse cosine of each side.  The inverse cosine and cosine undo each other, leaving us with just 3beta on the left, just like in the previous problems.  That simplifies to:

[tex]3\beta=cos^{-1}(\frac{1}{2})[/tex]

We look to the unit circle to find the values of beta that give us the cosine of 1/2 and those are:

[tex]3\beta =\frac{\pi}{6},3\beta  =\frac{5\pi}{6}[/tex]

Divide each of those by 3 to find the values of beta are:

[tex]\beta =\frac{\pi }{18} ,\frac{5\pi}{18}[/tex]

d.  [tex]sec3\alpha =-2[/tex]

Let's rewrite this in terms of a trig ratio that we are a bit more familiar with:

[tex]\frac{1}{cos(3\alpha) } =\frac{-2}{1}[/tex]

We are going to simplify this even further by flipping both fraction upside down to make it easier to solve:

[tex]cos(3\alpha)=-\frac{1}{2}[/tex]

Now we will take the inverse cos of each side (same as above):

[tex]3\alpha =cos^{-1}(-\frac{1}{2} )[/tex]

We look to the unit circle one last time to find the values of alpha that give us a cosine of -1/2:

[tex]3\alpha =\frac{7\pi}{6},3\alpha  =\frac{11\pi}{6}[/tex]

Dividing both of those equations by 3 gives us

[tex]\alpha =\frac{7\pi}{18},\frac{11\pi}{18}[/tex]

And we're done!!!

Answer:

Step-by-step explanation:

Let p represent the number of private-session tutoring students. Then (12-p) is the number of group-session students. Her total revenue is ...

  25p +18(12 -p) = 265

  7p + 216 = 265 . . . . . . . simplify

  7p = 49 . . . . . . . . . . . . . subtract 216

  p = 7 . . . . . . . . . . . . . . . . divide by 7

Lisa tutored 7 students privately and 5 in group session that day.

Answer:

32 degrees fahrenheit =

0 degrees celsius

Step-by-step explanation:

Formula

(32°F − 32) × 5/9 = 0°C

Answer:

  82.4 °F

Step-by-step explanation:

The appropriate conversion formula is ...

  F = 9/5C +32

For C = 28, this is ...

  F = (9/5)(28) +32 = 50.4 +32 = 82.4

The equivalent temperature in degrees Fahrenheit is 82.4.

[tex]A=\{a,e,i,o,u\}\\n(A)=5\\n(\mathcal{P}(A))=2^5=32\\\\x^2+3<2\\x^2<-1 \\x\in \emptyset\\|B|=0\\n(\mathcal{P}(B))=2^0=1[/tex]

The number of elements in the power set of A, where A consists of all vowels in the English alphabet, is 32, while for B, where B consists of natural numbers satisfying the condition x²+3<2 (which has no solutions), the power set has 1 element.

The student's question involves finding the number of power sets, denoted as n[p(A)] and n[p(B)], for two specific sets A and B. Set A is defined as {x; x is a vowel of the English alphabet}, which consists of 5 elements as there are 5 vowels in the English alphabet (A, E, I, O, U). The number of elements in a power set is given by 2 to the power of the number of elements in the original set, so for set A, the number of elements in the power set is 25 or 32. Therefore, n[p(A)] = 32.

Set B is defined as {x; x²+3<2, x is a natural number, denoted as N}. Solving the inequality x²+3<2, we find that no natural number satisfies this condition since the smallest value would be when x=1, and 1²+3 equals 4, which is not less than 2. Since no elements satisfy this inequality, set B is an empty set. The power set of an empty set has just one element, the empty set itself, so n[p(B)] = 1.

The two lines start at the same point outside the circle, using the intersecting Secants theorem, When you multiply the two dims for each line together, they are equal

For A:

5 * (5+x) = 6 * (6+4)

25 + 5x = 60

5x = 35

x = 35/5

x = 7

For B:

5 * (5+ 3) = 4 * ( 4+x)

40 = 16 + 4x

4x = 24

x = 24/4

x = 6

Answer: Option D

300°

Step-by-step explanation:

we have the following equation

[tex]tanx+\sqrt{3}=0[/tex]

To solve the equation add [tex]-\sqrt{3}[/tex] on both sides of equality

[tex]tanx+\sqrt{3}-\sqrt{3}=-\sqrt{3}[/tex]

[tex]tanx=-\sqrt{3}[/tex]

We apply the inverse function [tex]tan^{-1}x[/tex]

[tex]x=tan^{-1}(-\sqrt{3})[/tex]

[tex]x=-60\°[/tex] or [tex]x=300\°[/tex]

The answer is the option D

Since contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive.

Equations are expressions separated by an equal sign. Given the expression below

5x = 10

In order to get the value of x, we will divide both sides by 2 to have:

5x/5 = 10/5

x = 10/5

x = 2

Since contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, hence the statement above is contrapositive.

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Answer:

10 model cars

Step-by-step explanation:

1. We must find the cost of 3 wheels, so we can add it to the cost of the other materials.

    .50 × 3 = 1.50

2. We will now add the cost of the other materials to the cost of 3 wheels. This will give us the total cost of each car they build.

    1.50 + 1.25 = 2.75

3. Now, we will divide 30 by 2.75. This is because dividing 30 by 2.75 will show us how many times 2.75 can go into 30, essentially how many times they can purchase the total materials needed to make a model car.

    30/2.75 = 10.9090909....

4. Lastly we will remove the decimal from the 10. This is because the .909090.... represents them purchasing about 90% of the materials they need instead of another whole one because they ran out of money.

    10 model cars can be made

The equation is C = 20t^2 + 135t + 3050

You are told the total number of cars sold is 15000.

Replace c with 15,000 and solve for t:

15000 = 20t^2 + 135t + 3050

Subtract 15000 from both sides:

0 = 20t^2 + 135t - 11950

Use the quadratic formula to solve for t.

In the quadratic formula -b +/-√(b^2-4(ac) / 2a

using the equation, a = 20, b = 135 and c = -11950

The formula becomes -135 +/- √(135^2 - 4(20*-11950) / (2*20)

t = 21.3 and -28.1

T has to be a positive number, so t = 21.3,

Now you are told t = 0 is 1998,

so now add 21.3 years to 1998

1998 + 21.3 = 2019.3

So in the year 2019 the number of cars will be 15000

Answer:

The year 2019.

Step-by-step explanation:

Plug 15,000 into the variable C:

15,000 = 20t^2 + 135t + 3050

20t^2 + 135t - 11,950 = 0.  Divide through by 4:

4t^2 + 27t -  2390 = 0.

t = [ (-27 +/- sqrt (27^2 - 4 * 4 * -2390)] / (2*4)

= 21.3, -28.05  ( we ignore the negative value).

So the number of cars will reach 15,000 in 1998 + 21 = 2019.

Answer: y int: (0,9)    x int: (3,0)

Step-by-step explanation:

In slope intercept form, the equation is y=-3x+9. In the formula y=mx+b, we know b is the y intercept, so our y int. is 9. To find our x intercept, we set y=0. So, 0=-3x+9=>3x=9=>x=3

Answer:

736 Newtons

Step-by-step explanation:

Given

Pressure = [tex]\frac{Force}{Area}[/tex]

Multiply both sides by Area

Area × Pressure = Force

Area = 2.3 × 1.6 = 3.68 m², hence

Force = 3.68 × 200 = 736 Newtons

Answer:

Option D.) x + 2x − 3 = 26

Step-by-step explanation:

Let

x ------> the length of Nate's car

y ------> the length of Maya's car

we know that

x+y=26 -----> equation A

y=2x-3 ----> equation B

substitute equation B in equation A and solve for x

x+(2x-3)=26

3x=26+3

x=29/3 in

Find the value of y

y=2(29/3)-3

y=(58/3)-3

y= 49/3 in

Answer:

D.) x + 2x − 3 = 26

Step-by-step explanation:

took the test and got it right 100%. if you are wondering if this is the right answer for the test. if this is for homework look at the other persons answer. he gives it more in depth.hope this helped.

Answer:

$6750 in the bank account and $20,250 in the stock fund

Step-by-step explanation:

If B is the money they put in the bank and S is the amount they put in the stock fund, then:

B + S = 27000

1.024 B + 1.072 S = 1.06 × 27000

Solving the system of equations:

1.024 (27000 − S) + 1.072 S = 28620

27648 − 1.024 S + 1.072 S = 28620

0.048 S = 972

S = 20250

B = 27000 − S

B = 6750

They should put $6750 in the bank account and $20,250 in the stock fund.

Answer:

Amplitude = 3 feet; period = 8 hours; midline: y = 4

Step-by-step explanation:

sketch it....(see attached)

Answer:

The correct option is 3.

Step-by-step explanation:

It is given that the San Francisco Bay tides vary between 1 foot and 7 feet.

It means the maximum value of the function is 7 and minimum value is 1.

The amplitude of the function is

[tex]Amplitude=\frac{Maximum-Minimum}{2}[/tex]

[tex]Amplitude=\frac{7-1}{2}=\frac{6}{2}=3[/tex]

The amplitude of the function is 3 feet.

Midline of the function is

[tex]Midline=\frac{Maximum+Minimum}{2}[/tex]

[tex]Midline=\frac{7+1}{2}=\frac{8}{2}=4[/tex]

The midline of the function is 4 feet.

It is given that the tide completes a full cycle in 8 hours. It means the period of function is 8 hours.

Therefore the correct option is 3.

Answer:

  (a) 720 -2x

  (b) 0 ≤ x ≤ 360

  (c) A = x(720 -2x)

  (d) (30.334, 68.645) ∪ (291.355, 329.666) (two disjoint intervals)

  (e) x = 180 ft, the other side = 360 ft; total area 64,800 ft²

Step-by-step explanation:

(a) The two parallel sides of the fenced area are each x feet, so the remaining amount of fence available for the third side is (720 -2x) ft. Then ...

  length = 720 -2x

__

(b) The two parallel sides cannot be negative, and they cannot exceed half the length of the fence available, so ...

  0 ≤ x ≤ 360

__

(c) Area is the product of the length (720-2x) and the width (x). The desired function is ...

  A = x(720 -2x)

__

(d) For an area of 20,000 ft², the values of x will be ...

  20000 = x(720 -2x)

  2x² -720x +20000 = 0

  x = (-(-720) ±√((-720)² -4(2)(20000)))/(2(2)) = (720±√358400)/4

  x = 180 ±40√14 = {30.334, 329.666} . . . feet

For an area of 40,000 ft², the values of x will be ...

  x = 180 ±20√31 ≈ {68.645, 291.355} . . . feet

The values of x producing areas between 20,000 and 40,000 ft² will be values of x in the intervals (30.334, 68.645) or (291.355, 329.666) feet.

__

(e) The vertex of the area function is at the axis of symmetry: x = 180. The corresponding dimensions are ...

  180 ft × 360 ft

and the area of that is 64,800 ft².

The length of the remaining side to be fenced is 4x - 720 ft. The restrictions on x are that it must be greater than 180 ft. The area function A(x) is (4x - 720) * x. The values of x that give an area between 20,000 and 40,000 ft2 are 30 ft to 42 ft. The dimensions that give a maximum area are 42 ft by 42 ft, with an area of 17,640 ft2.

(a) Express the length of the remaining side to be fenced in terms of x:

The perimeter of a rectangle is the sum of all its sides. Since we know the width is x and there are two parallel sides of length x, we can express the perimeter as 2x + x + x = 4x. The remaining side to be fenced can be expressed as 4x - 720 ft.

(b) What are the restrictions on x:

The length of each side, x, cannot be negative or zero since it represents a physical length. Additionally, the remaining side to be fenced must be positive, so 4x - 720 > 0. Combining these restrictions, x > 180 ft.

(c) Determine a function A that represents the area of the parking lot in terms of x:

The area of a rectangle is given by length multiplied by width. In this case, the length is the remaining side to be fenced, so the function A representing the area is A(x) = (4x - 720) * x.

(d) Determine the values of x that will give an area between 20,000 and 40,000 ft2:

To find the values of x that give an area between 20,000 and 40,000 ft2, we can set A(x) between those values and solve for x. We get the inequality 20,000 ≤ (4x - 720) * x ≤ 40,000. Solving this inequality, we find that 30 ft ≤ x ≤ 42 ft.

(e) What dimensions will give a maximum area, and what will this area be:

To find the dimensions that will give a maximum area, we can maximize the area function A(x). We can do this by finding the critical points of A(x) by taking its derivative and setting it equal to zero. After solving this equation, we find that x = 30 ft and x = 42 ft are the critical points. Evaluating A(x) at these critical points, we find that the dimensions that give a maximum area are 42 ft by 42 ft, with an area of 17,640 ft2.

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Final answer:

The answer explains how to calculate the probability of not receiving one's own ID card using the inclusion-exclusion principle and provides the limit of this probability as n approaches infinity.

Explanation:

Inclusion-Exclusion Principle:

(a) To calculate the probability that no one receives their own ID card back, we use the principle of inclusion-exclusion. The probability is given by 1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n * 1/n!.

(b) As n approaches infinity, the probability approaches e-1 which is approximately 0.3679.

Answer:

(4,0)

Step-by-step explanation:

The object is first at (0,0)

It is reflected across line x=-2, this means you draw the mirror line at x=-2 and count 2 equal units backwards to get the image.The image will be at ;[tex]y=0\\\\x=-2-2=-4\\\\\\=(-4,0)[/tex]

The image (-4,0) is then reflected on the y-axis

You know reflection on the y-axis, the y-coordinate remains the same but the x-coordinate is changed to its opposite sign.

Hence;

(- -4,0)= (4,0)

The image will move 8 units towards positive x-axis.This is the same as moving 4 units from the mirror line at (0,0) and land at (4,0)

Answer:

Let X be the event of feeling secure after saving $50,000,

Given,

The probability of feeling secure after saving $50,000, p = 40 % = 0.4,

So, the probability of not  feeling secure after saving $50,000, q = 1 - p = 0.6,

Since, the binomial distribution formula,

[tex]P(x=r)=^nC_r p^r q^{n-r}[/tex]

Where, [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]

If 8 households choose randomly,

That is, n = 8

(a) the probability of the number that say they would feel secure is exactly 5

[tex]P(X=5)=^8C_5 (0.4)^5 (0.6)^{8-5}[/tex]

[tex]=56(0.4)^5 (0.6)^3[/tex]

[tex]=0.12386304[/tex]

(b) the probability of the number that say they would feel secure is more than five

[tex]P(X>5) = P(X=6)+ P(X=7) + P(X=8)[/tex]

[tex]=^8C_6 (0.4)^6 (0.6)^{8-6}+^8C_7 (0.4)^7 (0.6)^{8-7}+^8C_8 (0.4)^8 (0.6)^{8-8}[/tex]

[tex]=28(0.4)^6 (0.6)^2 +8(0.4)^7(0.6)+(0.4)^8[/tex]

[tex]=0.04980736[/tex]

(c) the probability of the number that say they would feel secure is at most five

[tex]P(X\leq 5) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)[/tex]

[tex]=^8C_0 (0.4)^0(0.6)^{8-0}+^8C_1(0.4)^1(0.6)^{8-1}+^8C_2 (0.4)^2 (0.6)^{8-2}+8C_3 (0.4)^3 (0.6)^{8-3}+8C_4 (0.4)^4 (0.6)^{8-4}+8C_5(0.4)^5 (0.6)^{8-5}[/tex]

[tex]=0.6^8+8(0.4)(0.6)^7+28(0.4)^2(0.6)^6+56(0.4)^3(0.6)^5+70(0.4)^4(0.6)^4+56(0.4)^5(0.6)^3[/tex]

[tex]=0.95019264[/tex]

Answer:

30 and 150

Step-by-step explanation:

Whether these are same side interior or same side exterior, the sum of them is 180 when they are on the same side of a transversal that cuts 2 parellel lines.  If angle A is 5 times smaller than angle B, then angle B is 5 times larger.  So angle A is "x" and angle B is "5x".  The sum of them is 180, so

x + 5x = 180 and

6x = 180 so

x = 30 and 5x is 5(30) = 150

Answer:

30 and 150

Step-by-step explanation:

YOUR WELCOME!

bearing in mind that perpendicular lines have negative reciprocal slopes, let's find firstly the slope of AC.

[tex]\bf (\stackrel{x_1}{2}~,~\stackrel{y_1}{1})\qquad (\stackrel{x_2}{1}~,~\stackrel{y_2}{6}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{6-1}{1-2}\implies \cfrac{5}{-1}\implies -5 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{perpendicular lines have \underline{negative reciprocal} slopes}} {\stackrel{slope}{\cfrac{-5}{1}}\qquad \qquad \qquad \stackrel{reciprocal}{\cfrac{1}{-5}}\qquad \stackrel{negative~reciprocal}{\cfrac{1}{5}}}[/tex]

so, we're really looking for the equation of a line whose slope is 1/5 and that passes through (3,3)

[tex]\bf (\stackrel{x_1}{3}~,~\stackrel{y_1}{3}) ~\hspace{10em}slope = m\implies \cfrac{1}{5} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-3=\cfrac{1}{5}(x-3) \implies y-3=\cfrac{1}{5}x-\cfrac{3}{5} \\\\\\ y=\cfrac{1}{5}x-\cfrac{3}{5}+3\implies y=\cfrac{1}{5}x+\cfrac{12}{5}[/tex]

Answer:

61 steps per minute

3,660 steps per hour

Step-by-step explanation:

To find how many steps she walks in a minute, you have to divide 1830 by 30.

So, 1830/30 = 61

So she takes 61 steps per minute.

She walks 1830 in 30 minutes, and there are 60 minutes in an hour. 30 is also half of 60, so you would multiply 1830 by 2 to find out how many steps she walks in an hour.

So, 1830*2 = 3,660

So she takes 3,660 steps per hour.

Answer:

At q=6.8 the revenue is maximum. So, q=6.8 thousand units.

Step-by-step explanation:

The revenue function is

[tex]R(q)=-q^3+140q[/tex]

where q is thousands of units and R ( q ) is thousands of dollars.

We need to find for what value of q is revenue maximized.

Differentiate the function with respect to q.

[tex]R'(q)=-3q^2+140[/tex]

Equate R'(q)=0, to find the critical values.

[tex]0=-3q^2+140[/tex]

[tex]3q^2=140[/tex]

Divide both sides by 3.

[tex]q^2=\frac{140}{3}[/tex]

Taking square root both the sides.

[tex]q=\pm \sqrt{\frac{140}{3}}[/tex]

[tex]q=\pm 6.8313[/tex]

[tex]q\approx \pm 6.8[/tex]

Find double derivative of the function.

[tex]R''(q)=-6q[/tex]

For q=-6.8, R''(q)>0 and q=6.8, R''(q)<0. So at q=6.8 revenue is maximum.

At q=6.8 the revenue is maximum. So, q=6.8 thousand units.

Answer:

A = 27π cm² or A ≈ 84.823... cm²

Step-by-step explanation:

From being shown that the missing area is 90 out of 360°, we know that we need to find 75% of the area of the given circle.

Our equation is altered to: A = 0.75πr²

Plug in: A = 0.75π(6)²

Multiply: A = 27π cm²

If you were instructed to multiply pi and round (which is not as probable given that this is RSM), then the answer would be A ≈ 84.823... cm²

Answer:

Exact answer: [tex]27\pi[/tex]

Answer rounded to nearest hundredths: 84.82 using the pi button and not 3.14.

Step-by-step explanation:

Let's pretend for a second the whole circle is there.

The area of the circle would be [tex]A=\pi r^2[/tex] where [tex]r=6[/tex] since 6 cm is the length of the radius.

So the area of the full circle would have been [tex]A=\pi \cdot 6^2[/tex].

[tex]A=\pi \cdot 6^2[/tex]

Simplifying the 6^2 part gives us:

[tex]A=\pi \cdot 36[/tex]

or

[tex]A=36 \pi[/tex]

Now you actually have one-fourth (because of the 90 degree angle located at the central angle) of the circle missing so our answer is three-fourths of what we got from finding the area of the full circle.

So finding three-fourths of our answer means taking the [tex]36\pi[/tex] we got earlier and multiplying it by 3/4.

This means the answer is [tex]\frac{3}{4} \cdot 36\pi[/tex].

3/4 (36)=3(9)=27

So the answer is [tex]27\pi[/tex]