Answers
Result:
sin(tan⁻¹(x) - tan⁻¹(y)) in terms of x and y only = (x - y) / (1 + xy).
How do we express the equation in term of x and y?For us to write the expression, we shall use this formular formula:
tan(a - b) = (tan(a) - tan(b)) / (1 + tan(a)tan(b))
where:
a = tan⁻¹(x)
b = tan⁻¹(y)
Next, we have:
tan(tan⁻¹(x) - tan⁻¹(y)) = (tan(tan⁻¹(x)) - tan(tan⁻¹(y))) / (1 + tan(tan⁻¹(x))tan(tan⁻¹(y)))
= (x - y) / (1 + xy)
From both sides, we have sine:
sin(tan⁻¹(x) - tan⁻¹(y)) = sin(tan(x - y / 1 + xy))
So, sin(tan⁻¹(x) - tan⁻¹(y))
= sin(tan⁻¹(x) - tan⁻¹(y))
= (x - y) / (1 + xy) in terms of x and y.
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Related Questions
I need help with this problem
Answers
Answer:
0.375
Step-by-step explanation:
Use the given circumference to find the surface area of the spherical object. a pincushion with C = 24 cm
Answers
Answer:
approximately 183.4 square centimeters.
Step-by-step explanation:
We can use the formula for the circumference of a sphere, which is:
C = 2πr
where C is the circumference and r is the radius.
We know that the circumference of the pincushion is 24 cm. We can use this to find the radius:
C = 2πr
24 = 2πr
r = 24/(2π)
r ≈ 3.82 cm
Now that we know the radius, we can use the formula for the surface area of a sphere, which is:
A = 4πr^2
Substituting the value we found for the radius, we get:
A = 4π(3.82)^2
A ≈ 183.4 cm^2
Therefore, the surface area of the pincushion is approximately 183.4 square centimeters.
Abbie needs to cover square pyramids with fabric. The dimensions of each pyramid are shown in the table. If the fabric costs $0.001 per square millimeter, what is the surface area and cost to cover each pyramid, rounded to the nearest whole cent?
Please help!!!!
Answers
The surface area of each square pyramid, we used the formula Surface Area = Base Area + 4 x (1/2 x Base x Height of Face) is 1049.2mm², 1285.8mm² and 2100mm². The cost to cover each pyramid was $1.05, $1.29 and $2.10.
To calculate the surface area of each pyramid, we need to use the formula
Surface Area = Base x Height of face + 2 x ((Base/2) x Slant Height)
For the small pyramid, we have
Surface Area = 20 x 25 + 2 x ((20/2) x 26.46)
Surface Area = 500 + 2 x 10 x 26.46
Surface Area = 1049.2 mm²
Cost to cover the small pyramid
Cost = Surface Area x 0.001
Cost = 1049.2 x 0.001
Cost = $1.05
For the medium pyramid, we have
Surface Area = 20 x 40 + 2 x ((20/2) x 42.43)
Surface Area = 800 + 2 x 10 x 42.43
Surface Area = 1285.8 mm²
Cost to cover the medium pyramid
Cost = Surface Area x 0.001
Cost = 1285.8 x 0.001
Cost = $1.29
For the large pyramid, we have
Surface Area = 30 x 40 + 2 x ((30/2) x 50)
Surface Area = 1200 + 2 x 15 x 50
Surface Area = 2100 mm²
Cost to cover the large pyramid
Cost = Surface Area x 0.001
Cost = 2100 x 0.001
Cost = $2.10
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Solve for x assume that lines which appear to be diameters are actual diameters
Answers
The value of x in the circle that shows diameter is calculated as:
x = -7
What is the Diameter of a Circle?In a circle, the line segment that divides the circle into two halves is the diameter. Half of the circle is a semicircle which has a measure of 180 degrees.
Note: The measure of a central angle is also the same measure with that of the arc it intercepts.
Therefore, we have:
-8x + 9 + 50 + 65 = 180
-8x + 124 = 180
-8x = 180 - 124
-8x = 56
x = 56/-8
x = -7
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What are the coordinates of the image of the point (8,1) after a dilation by a scale factor of 2 with the origin as the center of dilation, followed by a translation over the y-axis
Answers
The point after a dilation by a scale factor of 2 with the origin as the center of dilation, and a translation over the y-axis is (-16, 2).
Given is a point, (8, 1) we need to find the coordinates of a point which is dilated from the origin by the scale factor of 2 and translation over the y-axis,
So, the coordinates of after dilation by k scale factor =
(x, y) = (kx, ky)
So, (8, 1) = (16, 2)
Now, the translation over the y-axis gives,
(x, y) = (-x, y)
Therefore, (16, 2) = (-16, 2)
Hence, the point after a dilation by a scale factor of 2 with the origin as the center of dilation, and a translation over the y-axis is (-16, 2).
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can someone help me pls it Ricardo has made 4 out of the 12 recipes in his cookbook. What fraction of the recipe has he made? and thank you
Answers
Yes, of course!
Ricardo has made 4 out of the 12 recipes in his cookbook.
To find out what fraction of the recipes he has made, you can write it as a fraction:
4/12
To simplify this fraction, you can divide both the numerator and denominator by 4, which gives:
4/12 = 1/3
Therefore, Ricardo has made 1/3 of the recipes in his cookbook.
In △ABC, we are told that a=17, ∡B=70∘, and ∡C=48∘. Solve for b and c.
Answers
Answer: b=18.1 and c=14.3
Step-by-step explanation:o solve for the lengths of the sides b and c in triangle ABC, we can use the law of sines, which relates the lengths of the sides of a triangle to the sines of the angles opposite those sides. Specifically, the law of sines states that:
a/sin(A) = b/sin(B) = c/sin(C)
where A, B, and C are the angles of the triangle opposite sides a, b, and c, respectively.
Given that a = 17, B = 70°, and C = 48°, we can write:
b/sin(70°) = 17/sin(A)
c/sin(48°) = 17/sin(A)
To solve for b and c, we need to find sin(A). We can do this by using the fact that the angles of a triangle sum to 180°:
A + B + C = 180°
A = 180° - B - C
A = 180° - 70° - 48°
A = 62°
Now we can substitute sin(A) = sin(62°) into the equations above and solve for b and c:
48 inches by 36 inches what is the square feet
Answers
Answer:
1728 square ft²
Step-by-step explanation:
48x36=1728
Answer:
[tex]\large \boxed{\mathrm{Area}}[/tex] = [tex]\large \boxed{\mathrm{12 \ ft^2}}[/tex]
Steps:
12 inches = 1 foot
48 / 12 = 4 [tex]\meduim \boxed{\mathrm{feet}}[/tex]
36 / 12 = 3 [tex]\large \boxed{\mathrm{feet}}[/tex]
Answer:
3 x 4 = 12 ft²
Everyone should get 15 points that's how much I'm using i don't know if it's 100% correct.
Answers
The box plot that has the smallest interquartile range would be the fourth box plot.
The bar graph that shows Ellen's steps would be B. Second bar graph.
How to find the box plot and bar graph?The interquartile range on a box plot can be found by subtracting the figure where the box starts at, from the figure where the box plot stops.
This means that the figure with the least interquartile range is the fourth box plot with an interquartile range of :
= 8 - 4
= 4
The bar graph that would show Ellen's steps would be the one that has the most steps on the graph, closest to the mean of 10, 000 steps. Only Graph B has steps in the number of days, which are closest to 10, 000 steps consistently so this is the correct graph.
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3. Olivia's mother surprised her with pasta packed in a rectangular prism shaped lunch box. How much quantity of pasta is served to Olivia given that the dimensions of the rectangular prism are as follows:
length= 5 in , width= 4 in, height= 3 in
Answers
The quantity of pasta is served to Olivia is 60 in³.
Olivia's mother packed her lunch in a lunch box which is in the shape of rectangular prism, with dimension = length= 5 in, width= 4 in, height= 3 in,
We need to find the quantity of the lunched packed in the box,
So, we will find the volume of the box = length × width × height
= 5 × 4 × 3
= 60 in³
Hence the quantity of pasta is served to Olivia is 60 in³.
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Describe by a write up on how models can be used to express numbers of different bases to base 2 and base 5
Answers
Conversion models can be used to express numbers of different bases to base-2 and base-5. To convert a number from base-10 to base-2, we use division-by-2, and for base-5, we divide by 5. To convert between bases, we convert to base-10 first and then to the desired base. for example 25 in binary is 11001, 45 in base-5 is 104, 89 to base-2 is 1011001.
In computer science and mathematics, binary (base-2) and quinary (base-5) numbering systems are commonly used to represent numbers. However, it is often necessary to express numbers in other bases as well. This is where conversion models come into play.
To convert a number from base-10 to base-2, we can use the division-by-2 method.
The process involves repeatedly dividing the decimal number by 2 and keeping track of the remainders until the quotient becomes 0. We then write the remainders in reverse order to get the binary representation of the number.
For example, to convert the number 25 to binary
25 / 2 = 12 R1
12 / 2 = 6 R0
6 / 2 = 3 R0
3 / 2 = 1 R1
1 / 2 = 0 R1
Therefore, 25 in binary is 11001.
To convert a number from base-10 to base-5, we can use a similar method. Instead of dividing by 2, we divide by 5 and keep track of the remainders. For example, to convert the number 45 to base-5
45 / 5 = 9 R0
9 / 5 = 1 R4
1 / 5 = 0 R1
Therefore, 45 in base-5 is 104.
We can also convert a number from base-2 to base-5 or vice versa. To convert a number from base-2 to base-5, we can first convert it to base-10 and then to base-5. For example, to convert the binary number 101110 to base-5
1*2⁵ + 0*2⁴ + 1*2³ + 1*2² + 1*2¹ + 0*2⁰ = 46
46 to base-5 is 141
To convert a number from base-5 to base-2, we can first convert it to base-10 and then to base-2. For example, to convert the quinary number 324 to base-2
3*5² + 2*5¹ + 4*5⁰ = 89
89 to base-2 is 1011001
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LAST QUESTION FOR THE NIGHT I NEED HELP WITH IT I’M KINDA STRUGGLING HERE
Answers
The solution to the system of equations in this problem is given as follows:
(-1, -5).
How to solve the system of equations?The system of equations in the context of this problem is defined as follows:
y = x - 4.y = -3x - 8.The x-coordinate of the solution in the context of this problem is obtained equaling the two equations, as follows:
x - 4 = -3x - 8
4x = -4
x = -1.
Hence the y-coordinate of the solution is obtained as follows:
y = -1 - 4
y = -5.
Thus the solution is:
(-1, -5).
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if $5700 is invested in a savings account for which interest is compounded annually, and if the $5700 turns into $6100 in 12 years, what is the interest rate of the savings account?
Answers
Answer:
We can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where A is the ending amount, P is the principal (starting amount), r is the interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time in years.
We know P = $5700, A = $6100, n = 1 (since the interest is compounded annually), and t = 12. We can solve for r:
$6100 = $5700(1 + r/1)^(1*12)
$6100/$5700 = (1 + r)^12
1.0702 = (1 + r)^12
log(1.0702) = log[(1 + r)^12]
0.0291 = 12log(1 + r)
0.00243 = log(1 + r)
10^(0.00243) = 1 + r
r = 0.0059 or 0.59%
Therefore, the interest rate of the savings account is 0.59%.
please help me i relly need help
Answers
Answer:
h(x) = 3·2^x
Step-by-step explanation:
You want the equation of the exponential curve h(x) = a·b^x when it goes through the points (0, 3) and (1, 6).
PointsWhen x=0, the equation is ...
h(0) = a·b^0 = a
We know that h(0) = 3, so a = 3.
When x=1, the equation is ...
h(1) = 3·b^1 = 3b
We know that h(1) = 6, so 3b = 6, or b = 2.
The equation is h(x) = 3·2^x.
<95141404393>
Evaluate 6×[2−(4)×(–8)]
Answers
Answer:204
Step-by-step explanation:
pemdas
6x[2--32]=6×[34]=204
A different sample box has a height that is twice as long as the original box described in Problem 1. What is the
volume of this sample box? How does the volume of this sample box compare to the volume of the sample box ins
Problem 1
Answers
The original sample box had a volume of 343 cubic cm, which is half the volume of the new sample box.
Since the height of the new sample box is twice as long as the original box, its height will be 2×7=14 cm.
The length and width remain the same at 7 cm.
The volume of the new sample box is given by the formula:
V = lwh
Substituting the values, we get:
V = (7)(7)(14) = 686 cubic cm
Therefore, the volume of the new sample box is 686 cubic cm.
Comparing the volumes of the two boxes, we see that the new sample box has a larger volume than the original sample box.
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Krell Industries has a share price of $22.08 today. If Krell is expected to pay a dividend of $0.98 this year, and its stock price is expected to grow to $24.98 at the end of the year, what is Krell's dividend yield and equity cost of capital?
what is Krell's equity cost of capital, or expected total return ?
Answers
Krell's equity cost of capital, or expected total return, is 5.57%.
We have,
The dividend yield is the annual dividend divided by the current share price:
Dividend yield
= Annual dividend / Share price
= $0.98 / $22.08 = 0.0444 = 4.44%
The equity cost of capital is the expected total return on the stock, which includes both the dividend yield and the expected capital appreciation:
Equity cost of capital = Dividend yield + Capital Appreciation
= 4.44% + [(24.98 - 22.08) / 22.08]
= 4.44% + 0.1314
= 5.57%
Therefore,
Krell's equity cost of capital, or expected total return, is 5.57%.
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The table below show the data to find an exponential model.
x 1 4 7 8 10
y. 798 1078 1519 2075 3102
1 goes with 798, 4 goes with 1078, 7 goes with 1519, 8 goes with 2075, and 10 goes with 3102
a. Write the data set to be used in order to linearize the data.
b. Write the system of equations
c. Write the matrices.
d. Indicate the c values
e. Write the exponential model in the form Y=C* 10^bx
Answers
The data set to be used to linearize the data is:
x ln(y)
1 ln(798)
4 ln(1078)
7 ln(1519)
8 ln(2075)
10 ln(3102)
b. The system of equations is:
$\begin{pmatrix}5 & \sum_{i=1}^{5} x_i \\ \sum_{i=1}^{5} x_i & \sum_{i=1}^{5} x_i^2 \\\end{pmatrix}\begin{pmatrix}b \\ ln(C) \\\end{pmatrix}=\begin{pmatrix}\sum_{i=1}^{5} ln(y_i) \\ \sum_{i=1}^{5} x_i ln(y_i) \\\end{pmatrix}$
c. The matrices are:
$\begin{pmatrix}5 & \sum_{i=1}^{5} x_i \\ \sum_{i=1}^{5} x_i & \sum_{i=1}^{5} x_i^2 \\\end{pmatrix}\begin{pmatrix}b \\ ln(C) \\\end{pmatrix}=\begin{pmatrix}\sum_{i=1}^{5} ln(y_i) \\ \sum_{i=1}^{5} x_i ln(y_i) \\\end{pmatrix}$
$\begin{pmatrix}5 & 30 \\ 30 & 266 \\\end{pmatrix}\begin{pmatrix}b \\ ln(C) \\\end{pmatrix}=\begin{pmatrix}7.986 \\ 27.214 \\\end{pmatrix}$
d. Solving for b and ln(C) using a calculator or a matrix-solving tool, we get:
b = 0.2695
ln(C) = 6.6195
Taking exponential of ln(C) we get C = 760.9305.
Therefore, the values of c are: C = 760.9305.
e. The exponential model in the form Y=C * 10^(bx) is:
Y = 760.9305 * 10^(0.2695x).
birth weights for a simple random sample of 195 boys were recorded. the mean weight calculated for this sample was 3.04 kilograms and the standard deviation was 0.71 kilograms
Answers
Answer:
The sample mean birth weight for the 195 boys is 3.04 kilograms and the sample standard deviation is 0.71 kilograms.
Step-by-step explanation:
10. Find the surface area of the
polyhedron that can be
assembled from this net. Show
your reasoning.
Answers
The surface area of the polyhedron is calculated be equal to 224 square inches.
How to evaluate for the area of the polyhedronThe polyhedron can be observed to have a large rectangle between two identical triangles by the sides
area of rectangle = length × width
The large rectangle have;
length = (12 + 10 + 10) in = 32 in
width = 4 in
area of large rectangle the = 32 in × 4 in = 128 in²
area of one identical triangles = 1/2 × 12 in × 8 in = 48 in²
area of 2 identical triangles = 2 × 48 in² = 96 in²
Surface area of the polyhedron = 128 in² + 96 in²
Surface area of the polyhedron = 224 in²
Therefore, the surface area of the polyhedron is calculated be equal to 224 square inches.
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Find the value of the trig function indicated.
3)
tan 0
13
12
0pls help
Answers
The value of the identities are;
1. tan θ = 5/12
2. cot θ = √7/3
How to determine the identitiesTo determine the value of the identities, we need to take note of the different identities;
sinetangentcotangentsecantcosecantcosineFrom the information given, we need to take note that;
tan θ = opposite/adjacent
cot θ = inverse of tangent identity
cot θ = adjacent/opposite
then,
1. tan θ =
Opposite = 5
Adjacent = 12
tan θ = 5/12
2. cot θ = 4√7/12 = √7/3
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(+25)-5x5:2+2²
27-5×5=2+2²
Answers
Answer:
16.5 = 6
Step-by-step explanation:
The equation given is (+25)-5x5:2+2² = 27-5×5=2+2². When calculating, we first simplify the division to get (+25)-12.5+4 = 27-25+4. Then we add and subtract the values on either side to get 16.5 = 6. Finally, as the equation is incorrect, we can conclude that the initial assertion is false. It is important to carefully follow the order of operations and simplify expressions correctly to avoid errors in equations.
Find the ratio specified in the picture.
Find sin A.
A
12
13
C 5 B
Answers
The value of sin A using trigonometric ratios is: 12/13
How to solve trigonometric ratios?There are three main trigonometric ratios in right angle triangles and they are:
sin x = opposite/hypotenuse
cos x = adjacent/hypotenuse
tan x = opposite/adjacent
We want to find sin A and as such using trigonometric ratios on the attached right angle triangle, we have:
sin A = 12/13
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256 chairs in the cafeteria. this is 8 times as many chairs in the library. total chairs in the library
Answers
Answer:
32
Step-by-step explanation:
in the word problem “256 chairs in the cafeteria. this is 8 times the chairs in the library.” Which means 8 times the number of chairs in the library equals the number of chairs in the cafeteria.
L = library
8 x L = 256
to find L divided 256 by 8
256/8 = 32
L = 32
Check the answer
8 x 32 = 256
Calculate the value of x. Give your answer as an integer or as a fraction in its simplest form. 18 mm S 3 mm xmm T 10 mm8
Answers
The value of the missing length of the enlarged triangle in the simplest form would be = 60mm.
How to calculate the missing length of the triangle?To calculate the missing length, the formula for scale factor should be used such as follows;
Scale factor = Larger dimensions/smaller dimensions
length of ∆T = 10mm
length of ∆S = 3mm
scale factor = 10/3 = 3.33
If the width of ∆S = 18mm
width of ∆ T = 18×3.33
= 59.94 = 60mm
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The Sky Train from the terminal to the rental car and long term parking center is supposed to arrive every 8 minutes. The waiting times for the train are known to follow a uniform distribution.
What is the probability of waiting less than 2 minutes and more than 6 minutes?
Answers
The 30th percentile in the uniform distribution is 2.4 min.
The correct option is (b)
Probability :The probability formula is defined as the ratio of favorable outcomes to the ratio of total outcomes. For any event (E), :
P(E)= Number of favorable outcomes/ Number of total outcomes
[tex]\mu=\frac{a +b}{2}[/tex]
[tex]\sigma=\sqrt{\frac{(b-a)^2}{12} }[/tex]
P(x) = 1/ b - a if a < x < b
P(X < a) = 0
and
P(X > b) = 0
In this case, a = 0 and b = 8
The 30th percentile in the uniform distribution, this means that the probability is 0.30. Therefore:
0.3 = [tex]\frac{1}{8-0}.x[/tex]
0.3 (8 -0) = x
2.4 = x
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The given question is incomplete, complete question is :
The Sky Train from the terminal to the rental-car and long-term parking center is supposed to arrive every eight minutes. The waiting times for the train are known to follow a uniform distribution. Find the 30th percentile for the waiting times (in minutes).
a. 2
b. 2.4
c. 2.75
d. 3
As seen in the diagram below, Arun is building a walkway with a width of x feet to go around a swimming pool that measures 15 feet by 7 feet. If the total area of the pool and the walkway will be 713 square feet, how wide should the walkway be?
Answers
The walkway around the swimming pool should be 8 feet wide.
Calculating the width of the walkwayFrom the question, we are given:
Length of the pool (l) = 15 feet
Width of the pool (w) = 7 feet
width of walkway (X) = X feet
Total Area = Area of pool and walkway = 713 square feet
Let's derive equation for the pool and walkaway combined:
length of the pool and walkway = 15 + 2X [walkway is on both sides]
width of the pool and walkway = 7 + 2X
Total Area = (length x width) of the pool and walkway
713 = (15 + 2X) x (7 + 2X)
713 = 15(7 + 2X) + 2X(7 + 2X)
713 = 105 + 30X + 14X + 4X²
713 = 105 + 44X + 4X²
Rearrange the equation
4X² + 44X + 105 = 713
Collect like terms and express in a proper quadratic equation
4X² + 44X - 608 = 0
Dividing both sides by 4, we get
X² + 11X - 152 = 0
Solve this equation quadratically
X = [tex]\frac{-b \± \sqrt{b^{2} - 4ac}}{2a}[/tex]
where
a = 1
b = 11
c = -152
Plug in the value into the equation
X = [tex]\frac{-11 \± \sqrt{11^{2} - 4(1)(-152)}}{2(1)}[/tex]
X = [tex]\frac{-11 \± \sqrt{121 + 608}}{2(1)}[/tex]
X = [tex]\frac{-11 \± \sqrt{729}}{2}[/tex]
X = [tex]\frac{-11 \± 27}{2}[/tex]
X = [tex]\frac{-11 + 27}{2}[/tex] or [tex]\frac{-11 - 27}{2}[/tex]
X = [tex]\frac{16}{2}[/tex] or [tex]\frac{-38}{2}[/tex]
X = 8 or -19
But since the width of the walkway cannot be negative, the only valid solution is therefore:
X = 8
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The low temperatures for four nights in December for a city are recorded below.
−4°, 1°, 3°, −2°
Which lists the temperatures in the correct order from coldest to warmest?
Answers
HELP PLEASEE!! The aquarium has 6 fewer yellow fish than green fish. 40% of the fish are yellow. How many green fish are in the aquarium? Show your work. Show all steps please.
Answers
Green fish: x
Yellow fish: x - 6
So, the total number of fish = x + (x - 6) = 2x - 6
40% of the fish are yellow, so:
0.4 (2x - 6) = x - 6
0.8x - 2.4 = x - 6
- 0.2x = - 3.6
x = 18
Thus, green fish = 18
Write a function of the form y= A sin (Bx-C)+D that has period 8, phase shift -2, and the range -12 ≤y≤-4.
y =
Answers
A function of the form y= A sin (Bx-C)+D that has period 8, phase shift -2, and the range -12 ≤y≤-4 is y = 4 sin (π/4 x + π/2) - 8
To write a function of the form y = A sin (Bx - C) + D that has a period of 8 and phase shift of -2, we can use the general formula:
y = A sin [(2π/P)(x - C)] + D
where P is the period, C is the phase shift, and D is the vertical shift. In this case, P = 8 and C = -2, so we have:
y = A sin [(2π/8)(x + 2)] + D
Simplifying the equation, we get:
y = A sin (π/4 x + π/2) + D
To find the amplitude A and vertical shift D that satisfy the range -12 ≤ y ≤ -4, we can use the fact that the sine function oscillates between -1 and 1. If we set A = 4, then the maximum value of y is 4 + D, and the minimum value of y is -4 + D. To ensure that the range is -12 ≤ y ≤ -4, we need to choose D such that:
4 + D ≤ -4 and -4 + D ≥ -12
Solving these inequalities, we get:
-8 ≤ D ≤ 0
Therefore, a function that satisfies the given conditions is:
y = 4 sin (π/4 x + π/2) - 8
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