Find the points of horizontal tangency to the polar curve. r = 4 csc θ + 5 0<θ < 2π . (r, θ )=_____ (smaller r-value) . (r, θ) = ____ (larger r-value)
Answers
The points of horizontal tangency on the polar curve r = 4csc(θ) + 5, where 0 < θ < 2π, are (9, π/2) (smaller r-value) and (1, 3π/2) (larger r-value).
To find the points of horizontal tangency to the polar curve given by r = 4csc(θ) + 5, where 0 < θ < 2π, we need to find the values of θ where the derivative of r with respect to θ is equal to zero.
First, let's express r in terms of θ using the trigonometric identity csc(θ) = 1/sin(θ):
r = 4csc(θ) + 5
r = 4/(sin(θ)) + 5
Now, let's find the derivative of r with respect to θ:
dr/dθ = d/dθ (4/(sin(θ)) + 5)
dr/dθ = -4cos(θ)/(sin²(θ))
To find the points of horizontal tangency, we need to solve the equation dr/dθ = 0. In this case, that means solving -4cos(θ)/(sin²(θ)) = 0.
Since the denominator sin²(θ) is never zero, the only way for the equation to be true is if the numerator -4cos(θ) is equal to zero. This occurs when cos(θ) = 0, which happens at θ = π/2 and θ = 3π/2.
Now, let's find the corresponding values of r at these angles:
For θ = π/2:
r = 4csc(π/2) + 5
r = 4(1) + 5
r = 9
For θ = 3π/2:
r = 4csc(3π/2) + 5
r = 4(-1) + 5
r = 1
Therefore, the points of horizontal tangency on the polar curve r = 4csc(θ) + 5, where 0 < θ < 2π, are (9, π/2) (smaller r-value) and (1, 3π/2) (larger r-value).
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Related Questions
Jenna won 77 toys playing basketball at her school's game night. Later, she gave four to each of her friends. She only has 9 remaining. How many friends does she have?
Answers
The equation can be written as:
77 - 4x = 9 where 77 is the total number of toys, 4 is how many toys she gave to each friend, x is the number of friends, and 9 is the toys remaining.
To solve for x, the number of friends:
77 -4x=9
Subtract 77 from both sides
77-77 -4x= 9-77
-4x= -68
Then divide both sides by -4 to isolate x.
-4x/-4 = 1x and -68/-4 = 17.
1x= 17, so the number of friends is 17.
you roll a fair six-sided die what is probability not three
Answers
How many ways can two teachers, four male students, five female students and one administrator be arranged if the teachers must sit together, the male students must sit together and the female students must sit together?
Answers
Using the Fundamental Counting Theorem, it is found that there are 160 ways for them to be arranged.
What is the Fundamental Counting Theorem?It is a theorem that states that if there are n things, each with \(n_1, n_2, \cdots, n_n\) ways to be done, each thing independent of the other, the number of ways they can be done is:
\(N = n_1 \times n_2 \times \cdots \times n_n\)
For this problem, we have that the parameters are found as follows:
There are four roles, hence \(n_1 = 4\).There are two teachers, hence \(n_2 = 2\).There are four male students, hence \(n_3 = 4\).There are five female students, hence \(n_4 = 5\).There is one administrator, hence \(n_5 = 1\).Thus the number of ways in which they can be arranged is:
N = 4 x 2 x 4 x 5 x 1 = 160.
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3. Marlene's meal at KFC costs there was a 10% service $110. How much will she pay altogether if charge and a 9% government tax on bills?
Answers
Marlene will have to pay a Total of $131.89 if there is a 9% government tax on bills.
Marlene had a meal at KFC which cost her $110, and there was an additional 10% service charge. We have to determine the total cost of her meal, including the 9% government tax on bills. To find out the total cost of the meal, we'll use the following steps:
Step 1: Calculate the service charge service Charge = 10% of $110Service Charge = (10/100) × $110Service Charge = $11
Step 2: Calculate the subtotal Subtotal = Cost of meal + Service charge Subtotal = $110 + $11Subtotal = $121
Step 3: Calculate the government tax Government Tax = 9% of Subtotal Government Tax = (9/100) × $121Government Tax = $10.89
Step 4: Calculate the total cost Total Cost = Subtotal + Government Tax Total Cost = $121 + $10.89Total Cost = $131.89
Therefore, Marlene will have to pay a total of $131.89 if there is a 9% government tax on bills.
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Which graph represents the solution set of the inequality x + 2 greater-than-or-equal-to 6 A number line goes from negative 9 to positive 9. A solid circle appears on positive 3. The number line is shaded from positive 3 through negative 9. A number line goes from negative 9 to positive 9. An open circle appears at positive 3. The number line is shaded from positive 3 through positive 9. A number line goes from negative 9 to positive 9. A closed circle appears at positive 4. The number line is shaded from positive 4 through positive 9. A number line goes from negative 9 to positive 9. An open circle appears at positive 4. The number line is shaded from positive 4 through negative 9.
Answers
Answer:
C. A number line goes from negative 9 to positive 9. A closed circle appears at positive 4. The number line is shaded from positive 4 through positive 9Step-by-step explanation:
Given inequality:
x + 2 ≥ 6Solution is:
x ≥ 6 - 2x ≥ 4Since the number line is restricted to -9, + 9 interval, the solution will appear as shaded part to the right from 4 through 9, where 4 is included point.
Correct answer choice is C
An upscale resort has built its circular swimming pool around a central area that contains a restaurant. The central area is a right triangle with legs of 60 feet, 120 feet, and approximately 103.92 feet. The vertices of the triangle are points on the circle. The hypotenuse of the triangle is the diameter of the circle. The center of the circle is a point on the hypotenuse (longest side) of the
Answers
The center of the circle, and consequently the central point of the resort's swimming pool, is located at the intersection of the two legs of the right triangle, approximately 60 feet from one vertex and 120 feet from the other.
The upscale resort has ingeniously designed its circular swimming pool to encompass a central area containing a restaurant. This central area takes the form of a right triangle with legs measuring 60 feet and 120 feet, while the hypotenuse, the longest side of the triangle, spans approximately 103.92 feet. The vertices of the triangle neatly coincide with points on the circumference of the circular pool.
Due to the properties of a right triangle, the hypotenuse is also the diameter of the circle. This means that the circular pool is precisely constructed around the right triangle, with its center located at the midpoint of the hypotenuse.
To determine the exact coordinates of the center of the circle, we can consider the properties of right triangles. Since the legs of the right triangle are perpendicular to each other, the midpoint of the hypotenuse coincides with the point where the two legs intersect.
In this case, the center of the circle is the point of intersection between the 60-foot leg and the 120-foot leg of the right triangle.
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What are the factors and zeros of the function f(x)=−2x2+7x+15?
Answers
Solving the quadratic function, it is found that the factors/zeros of the function are given by x = -1.5 and x = 5.
What are the factors/zeros of a function?They are given by the solutions of the function, that is, the values of x for which f(x) = 0.
In this problem, the function is given by:
f(x) = -2x² + 7x + 15.
Then:
-2x² + 7x + 15 = 0.
Which is a quadratic function with coefficients a = -2, b = 7, c = 15, hence:
\(\Delta = 7^2 - 4(-2)(15) = 169\)
\(x_1 = \frac{-7 + \sqrt{169}}{2(-2)} = -1.5\)
\(x_2 = \frac{-7 - \sqrt{169}}{2(-2)} = 5\)
The factors/zeros of the function are given by x = -1.5 and x = 5.
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What is the equation of the line?
A) y=34x−1
B)y=34x+1
C)y=43x−1
D)y=43x+1
Answers
Answer:
B
Step-by-step explanation:
not so sure but see if this is correct
Question 1 of 5 Select the correct answer from the drop-down menu. Triangle ABC represents a triangular garden with side lengths A B = 10 m , B C = 8 m , and A C = 15 m . What is the smallest and largest angle of the garden? The smallest angle is ∠ . The largest angle is ∠ .
Answers
Using cosine rule and sine rule, the angles In the triangular garden are
the smallest angle is 67.59 degreesthe largest angle is 38.94 degreesHow to find the smallest and the largest angleThe angles are found using the dimensions,
the angle opposite the smallest dimension gives the smallest anglethe angle opposite the largest dimension gives the largest angleThe angles are found first using cosine rule by the formula
cos c = (AC² - AB² - BC²) ÷ 2 * AB * BC
Solving for the largest angle
substituting the values
cos γ = (15² - 8² - 10²) ÷ 2 * 8 * 10
< γ = arc cos ( 0.38125 )
< γ = 67.59 degrees
Solving for the smallest angle
using sin rule
sin α / BC = sin γ / AC
sin α / 8 = sin 67.59 / 15
sin α = 8 * sin 67.59/15
α = arc sin 0.6285
α = 38.94 degrees
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I will give 100 points for 5 questions
Answers
Answer:
1. SSS
2. SAS
3. SAS
4. 1?
5. SSS
Step-by-step explanation:
40 PTS!!!! A single-serving juice boxes measure 10cm by 7cm and 5cm. A manufacturer wants to shrink wrap four boxes together for sale. Which of the two arrangements of the boxes with use the least amount of plastic wrap? Show your work
Answers
Both arrangements use the same amount of plastic wrap, with a total surface area of 1240 cm².
To determine which arrangement of the four juice boxes uses the least amount of plastic wrap, we need to consider the total surface area that needs to be covered in each arrangement.
Arrangement 1:
In this arrangement, we can place the four juice boxes in a 2x2 square configuration, with two boxes stacked horizontally and two boxes stacked vertically. Let's calculate the total surface area to be covered:
Surface area of a single box = 2(lw) + 2(lh) + 2(wh)
= 2(107) + 2(105) + 2(7*5)
= 140 + 100 + 70
= 310 cm²
Since we have four boxes in this arrangement, the total surface area to be covered is:
Total surface area = 4 * 310 cm²
= 1240 cm²
Arrangement 2:
In this arrangement, we can stack the four juice boxes in a single column, one on top of the other. Let's calculate the total surface area for this arrangement:
Surface area of a single box is the same as in the previous arrangement:
Surface area of a single box = 310 cm²
Since we have four boxes stacked vertically, the total surface area to be covered is:
Total surface area = 4 * 310 cm²
= 1240 cm²
Therefore, both arrangements use the same amount of plastic wrap, with a total surface area of 1240 cm².
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What is Van t Hoff law explain?
Answers
Van 't Hoff's law, also known as the law of osmotic pressure, states that the magnitude of the osmotic pressure is directly proportional to the concentration of solute particles in a solution at a given temperature.
Van 't Hoff's law is an important concept in chemistry and thermodynamics. It explains the relationship between osmotic pressure and the concentration of solute particles in a solution. According to the law, the osmotic pressure of a solution is directly proportional to the number of solute particles in the solution, regardless of the type of solute. This means that if the concentration of solute particles in a solution increases, the osmotic pressure also increases, and vice versa. This law has numerous applications, particularly in the field of biology, where it is used to explain the movement of water and nutrients in and out of cells. For example, when a plant cell is placed in a hypertonic solution, water moves out of the cell, causing it to shrink. On the other hand, when a plant cell is placed in a hypotonic solution, water moves into the cell, causing it to expand. Understanding Van 't Hoff's law is therefore critical in many fields of science, particularly in biology, chemistry, and chemical engineering.
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HP usually sells its laser printers at discounted prices and charges its customers much more when those customers buy consumables such as printer cartridges. What can we learn from this
Answers
HP's pricing strategy allows them to capture a larger customer base, create customer loyalty and long-term profitability.
HP uses the 'razor' and 'blade' model. Here, Laser printer is sold at a lower price while the consumables (printer cartridges) are sold at higher profit margins. This strategy aims to create a recurring revenue stream from the sales of consumables, which have higher profit margins compared to the initial printer sale.
By relying on the sales of consumables, HP diversifies its revenue streams. While printer sales may have lower profit margins, the higher profit margins from consumables help offset those costs and contribute to overall profitability. This reduces HP's reliance on a single revenue source and improves financial stability.
Therefore , HP is leaning towards a scheme which aims at long-term profitability.
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PLEASE HELP I DONT UNDERSTAND I’ll give u the brainliest thing
How do you solve quadratic equation using Completing the Square. Create your own example and state the steps.
Answers
Answer:
Now we can solve a Quadratic Equation in 5 steps:
Step 1 Divide all terms by a (the coefficient of x2).
Step 2 Move the number term (c/a) to the right side of the equation.
Step 3 Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.
Answer:
Example:
x^2+4x+3=0x^2+4x+4-1=0 here we add 1 to 3 and remove 1 to complete a square of (x+2) (x+2)^2 - 1=0 here we have square and 1 can be shown as 1^2using a^2-b^2= (a+b)(a-b) formula to factorize the polynomial(x+2)^2- 1^2=0(x+2+1)(x+2-1)=0(x+3)(x+1)=0x+3=0 and x+1=0now easily solved
which graph represents the system of inequalities
Answers
Answer:
The correct option is 2.
Step-by-step explanation:
The given system of inequalities
Inequality (1) can be written as
The related equation is
The slope of line is 5 and y-intercept is -2.
The sign of inequality is < , it means related line of inequality (1) is a dotted line and shaded region is below the line.
Inequality (2) can be written as
The related equation is
The slope of line is -3 and y-intercept is 4.
The sign of inequality is >, it means related line is a dotted line and shaded region is above the line.
Inequality (3) is
The related equation is
The slope of line is 2 and y-intercept is 0.
The sign of inequality is ≥, it means related line is a solid line and shaded region is above the line.
In the below graph the common shaded region is the solution region.
Therefore the correct option is 2.
Please help now asap !!!!!!!
Answers
Answer:
15=3×5
Step-by-step explanation:
i hope i helped
Thanks
Help please i need help please help me please
Answers
Answer:
quardent 3 please mark me brainlest
a carton of milk contains 1 1/2 servings of milk. a dozen cartons are poured into a large container and then poured into glasses that each hold 2/3 of a serving. how many glasses can be filled?
show work pls
Answers
Answer:
You can fill 27 glasses with milk
Step-by-step explanation:
Amount of servings, which is number of cartons times serving per carton, divided by the amount the glass can hold.
Let's solve the amount of servings:
Serving per carton: 1 1/2 is 3/2
Number of cartons: 12 (dozen)
12*3/2 = 18 servings
Now divide it by the amount the glass can hold:
18 ÷ 2/3 = 18*3/2 = 27 glasses
PLZ GIVE A REAL ANSWER Patsy the Peacock sells rare feathers. Below is the number of feathers she sold in each of the last eight months.
19,13,8,4,
7,14,7,9
Using the data, create a histogram.
Answers
Given data:
months no. of feathers
1 19
2 13
3 8
4 4
5 7
6 14
7 7
8 9
A Histogram is a graphical representation of a group of data points.
You can use Microsoft Excel to draw histograms.
Type the data set in the Excel sheet.Select the data.Go to Insert.Select chart type.The chart is displayed.Please see the attached images for reference.
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view link to get the answer
How many solutions does this equation have?
–6 − 8r = 8 − r
Answers
Answer:
1
Step-by-step explanation:
–6 − 8r = 8 − r
− 8r - r = 8 + 6
-9r = 14
r = -14/9
Answer:
r =-2
Step-by-step explanation:
–6 − 8r = 8 − r
Add r to each side
–6 − 8r+r = 8 − r+r
-6 -7r = 8
Add 6 to each side
-6+6 -7r = 8+6
-7r = 14
Divide each side by -7
-7r/-7 = 14/-7
r =-2
(57x -32< 72 respuesta son desigualdades
Answers
Answer:
x < 2
Step-by-step explanation:
57x - 32 < 72
57x < 104
x < 2
Kyle read 3 5/6 books and Jan read 2 1/3 books. How many more books did Kyle read than Jan?
Answers
Answer:
3 5/6 = 3 10/12
2 1/3= 2 4/12
3-2=1
10/12-4/12= 6/12=1/2
so Kyle read 1 1/2 more books then Jan
Answer:
Kyle read \(1 \frac{1}{2}\) books more than Jan
Step-by-step explanation:
Let's put our equation\(3 \frac{5}{6}-2\frac{1}{3}\)
They have different denominators so let's look for the \(LCM\) to find our common denominator
Find our LCMDefinition of LCM - LCM stands for least common multiple. The least common multiple of two numbers is the smallest number that is a multiple of both of them.
Now let's find our LCM
\(6:6,12,18,24,30,36,42\\\\3:3,6,9,12,15,18,21,24,27,30\)
as you can see our \(LCM\) is \(6\)
Now our fraction is\(3\frac{5}{6}-2\frac{2}{6}\)Solve\(3\frac{5}{6}-2\frac{2}{6}=1 \frac{3}{6}\) this can be reduced to \(1 \frac{1}{2}\)
Kyle read \(1 \frac{1}{2}\) books more than Jan
Brainliest please :)Write the slope intercept form of the equation of each line given the slope Y- intercept. Slope =9, y- intercept = -5.
Please help me
Answers
Answer:
y = 9x - 5
Step-by-step explanation:
Slope Intercept Form: y = mx + b
( m = slope and b = y-intercept form )
divide: 994.14 ÷ 18.9
Answers
Answer:
52.6?
Step-by-step explanation:
Step-by-step explanation:
\( \\ 994.14 \div 18.9\)
\( \\ = \frac{99414}{100} \div \frac{189}{10} \)
\( \\ = \frac{49707}{50} \div \frac{189}{10} \)
\( \\ = \frac{49707}{50} \times \frac{10}{189} \)
\( \\ = \frac{263}{5} \)
\( \\ = 52.6\)
-6(-4/7)=
I need for a math assignment please help
Answers
Answer:
The answer is 24/7
Answer:
3.428
Step-by-step explanation:
At a local fair, 194 balloons are blown up. over the course of the day, 85 of them are given away to children and another 38 of them pop. How many balloons are left at the end of the day?
Answers
Answer:
74
Step-by-step explanation:
What is the sum of 42÷9198?
Answers
Answer: 219
Step-by-step explanation: hope this helps!!!!
How do I find the area ?
Answers
Answer:
7.74 cm^2
Step-by-step explanation:
The length of the side of the square is 3+3 = 6
The area of the square is
A =s^2 = 6^2 = 36
The area of the circle is
A = pi r^2 where the radius is 3
(3.14) 3^2 =28.26
Subtract the area of the circle from the area of the square to get the shaded area
36-28.26 = 7.74 cm^2
Perform the indicated operation and simplify the result. 9x^(2)-1/12x^(2)-12x divide by 9x^(2)+12x+3/3x^(2)-6x+3 =
Answers
Given expression: \(\frac{9x^2 - 1}{12x^2 - 12x} \div \frac{9x^2 + 12x + 3}{3x^2 - 6x + 3}\) . We can simplify the given expression by multiplying the numerator and denominator of the first fraction by (3x-1) and then factorise. So, the simplified expression is \($\frac{3(x-1)^2}{4x(3x+1)(x+1)}$\).
In the second fraction, we can factorise the quadratic expression in the numerator.
= \(\frac{9x^2 - 1}{12x^2 - 12x} \cdot \frac{3x^2 - 6x + 3}{9x^2 + 12x + 3}\)
= \(\frac{(3x-1)(3x+1)}{12x(x-1)} \cdot \frac{3(x^2 - 2x + 1)}{3(3x^2 + 4x + 1)}\)
Simplify the expression.
= \(\frac{(3x-1)(x-1)}{4x(x-1)} \cdot \frac{(x-1)^2}{(3x+1)(x+1)}\)
= \(\frac{3(x-1)}{4x} \cdot \frac{(x-1)}{(3x+1)(x+1)}\)
= \(\frac{3(x-1)^2}{4x(3x+1)(x+1)}\) . Thus, the simplified expression is \($\frac{3(x-1)^2}{4x(3x+1)(x+1)}$\).
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David is making rice for his guests based on a recipe that requires rice, water, and a special blend of spice, where the rice-to-spice ratio is
15
:
1
15:115, colon, 1. He currently has
40
4040 grams of the spice blend, and he can go buy more if necessary. He wants to make
10
1010 servings, where each serving has
75
7575 grams of rice. Overall, David spends
4.50
4.504, point, 50 dollars on rice.
What is the price of rice per gram?
Answers
Answer:
The price of rice per gram is 4.50 / 750 = 0.006 dollars per gram.
Step-by-step explanation:
David wants to make 10 servings, where each serving has 75 grams of rice. So, he needs a total of 10 * 75 = 750 grams of rice. the price of rice per gram is 4.50 / 750 = 0.006 dollars per gram.