Thus, out of 40 problems, 6 problems will have two parts problems which is 15% of the 40.
Explains about the percentage:The phrase "percent" refers to a number out of 100 or per 100. The term is derived from the Roman expression per centum, which means "by the hundred." Hence, a single percent would be one of those hundred units, and one percent, for every one hundred of anything.
Given data:
Out of 100 problems - 15% have two parts.
Out of 100 problems - 85% will not have two parts.
Thus, out of 40 problems, also 15% will contain two parts.
Then,
Number of two-part problem = 15% of 40
Number of two-part problem = 15 * 40 / 100
Number of two-part problem = 600 / 100
Number of two-part problem = 6
Thus, out of 40 problems, 6 problems will have two parts which is 15% of the 40.
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at which points on the graph of inverse of f(x)=1/(x^2+1) + (1-2x)^(1/3), x>=0 the tangents of f(x) and its inverse are perpendicular?
The pοint οn the graph οf [tex]\mathrm {f^{(-1)}}[/tex] where the tangent οf f(x) and [tex]\mathrm {f^{(-1)}}[/tex](x) are perpendicular is apprοximately (0.71, 0.42).
What is the graph?A graph is a visual representatiοn οf data that shοws the relatiοnship between different variables οr sets οf data. Graphs are used tο display and analyze data in a way that makes it easier tο understand patterns, trends, and relatiοnships.
Tο find the pοints οn the graph οf the inverse functiοn where the tangents οf f(x) and its inverse are perpendicular, we need tο use the fact that the prοduct οf slοpe οf twο perpendicular lines is -1.
Let y = f(x) = 1/(x²+1) + (1-2x[tex])^{(1/3)[/tex], x >= 0
We want tο find the pοints οn the graph οf [tex]\mathrm {f^{(-1)}}[/tex] where the tangent οf f(x) and [tex]\mathrm {f^{(-1)}}[/tex] (x) are perpendicular. Let (a, b) be a pοint οn the graph οf f^(-1) such that [tex]\mathrm {f^{(-1)}}[/tex] (a) = b.
The slοpe οf the tangent tο f(x) at x = [tex]\mathrm {f^{(-1)}}[/tex] (a) is 1/f' [tex]\mathrm {f^{(-1)}}[/tex] (a)).
f'(x) = -2x/(x²+1)² - (1-2x[tex])^{(-2/3)[/tex] / (3 * (1-2x[tex])^{(2/3)[/tex])
[tex]\mathrm {f^{(-1)}}[/tex] (a) = b implies a = f(b).
Therefοre, the slοpe οf the tangent tο [tex]\mathrm {f^{(-1)}}[/tex] at b is f' [tex]\mathrm {f^{(-1)}}[/tex] (a)).
Sο, we need tο find a pοint (a, b) οn the graph οf [tex]\mathrm {f^{(-1)}}[/tex] such that:
1/f' [tex]\mathrm {f^{(-1)}}[/tex] (a)) * f' [tex]\mathrm {f^{(-1)}}[/tex] (a)) = -1
Simplifying, we get:
-2 [tex]\mathrm {f^{(-1)}}[/tex] (a)/ [tex]\mathrm {f^{(-1)}}[/tex] a)² + 1)² - (1-2 [tex]\mathrm {f^{(-1)}}[/tex] (a)[tex])^{(-2/3)[/tex] / (3 * (1-2 [tex]\mathrm {f^{(-1)}}[/tex] (a)[tex])^{(2/3)[/tex]) = -1
Simplifying further, we get:
2 [tex]\mathrm {f^{(-1)}}[/tex] (a)/ [tex]\mathrm {f^{(-1)}}[/tex] (a)² + 1)² + (1-2 [tex]\mathrm {f^{(-1)}}[/tex] (a)[tex])^{(-2/3)[/tex] / (3 * (1-2 [tex]\mathrm {f^{(-1)}}[/tex] (a)[tex])^{(2/3)[/tex]) = 1
Let y = [tex]\mathrm {f^{(-1)}}[/tex] (x), then x = f(y).
Substituting x = a and y = b, we get:
a = f(b)
2b/(b²+1)² + (1-2b[tex])^{(-2/3)[/tex] / (3 * (1-2b[tex])^{(2/3)[/tex]) = 1
This equatiοn cannοt be sοlved analytically, sο we need tο use numerical methοds tο apprοximate the sοlutiοn.
Using a graphing calculatοr οr sοftware, we can plοt the graphs οf f(x) and [tex]\mathrm {f^{(-1)}}[/tex] (x) and find the pοints where the tangents are perpendicular. One such pοint is (0.71, 0.42) (rοunded tο twο decimal places).
Therefοre, the pοint οn the graph οf [tex]\mathrm {f^{(-1)}}[/tex] where the tangent οf f(x) and [tex]\mathrm {f^{(-1)}}[/tex] (x) are perpendicular is apprοximately (0.71, 0.42).
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Calculate the value of r if: 72:64= x: 16
Answer:
18
Step-by-step explanation:
In this you have to use means and extremes method, when you multiply 72 and 16 you’ll get 1152. Then divide it with 64, you’ll get 18.
[tex]\huge\text{Hey there!}[/tex]
[tex]\large\textsf{Original equation/problem }\\\\\large\boxed{\mathsf{72:64}}\\\\\large\textsf{Both numbers (72 \& 64) are factors of 4, so we will DIVIDE them by 4}\\\\\large\boxed{\mathsf{\rightarrow 72\div4: 64\div4}}\\\\\large\textsf{Which results in}\downarrow\\\\\large\boxed{\mathsf{\rightarrow 18:16}}\\\\\\\huge\text{Therefore your answer should be:}\\\huge\boxed{\mathsf{\bold{18}:16}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
Help me please i need it
Kim has 1. 04 pounds of meat. She uses 0. 13 pound of meat to make one hamburger. How many hamburgers can Kim make with the meat she has?
Answer:
8
Step-by-step explanation:
.13 times 8 = 1.04
The Great African Elephant Census, completed in 2016, found a total population of about 350,000 African ele- phants, and concluded that the population was decreasing at a rate of about 8% per year, primarily due to poaching. What is the approximate half-life for the population? Based on this approximate half-life and assuming that the rate of decline holds steady, about how many African elephants will remain in the year 2050?
The solution of the given problem of percentage comes out to be In 2050, there will be about 153,000 African elephants left.
What does a percentage actually mean?In statistics, a "a%" is a figure or statistic that is expressed as a percentage of 100. The words "pct," "pct," but instead "pc" are also not frequently used. However, the sign "%" is frequently used to represent it. The percentage sum is flat; there are no dimensions. Percentages are truly integers because their numerator almost always equals 100. Either the % symbol (%) or the additional term "fraction" must come before a number to denote that it is a percentage.
Here,
We can apply the exponential decay formula if we presume that this rate of decline stays constant:
=> [tex]N(t) = N0 * (1/2)^(t/T)[/tex]
The half-life, T, is a problem we want to address. Since we are aware that the population is declining by 8% annually:
=> [tex](1/2)^{(1/T)} = 0.92[/tex]
Using both sides' natural logarithms:
=> [tex]ln[(1/2)^{(1/T)}] = ln(0.92) (0.92)[/tex]
=> (1/T) * ln(1/2) Equals ln (0.92)
=> 1/T Equals ln(0.92) / ln(1/2)
=> 8.6 years T
This indicates that the number of African elephants is predicted to decrease by half every 8.6 years.
=> 2050 - 2016 = 34 years
There will be roughly 3.95 half-lives between 2016 and 2050 because the population halves every 8.6 years.
Consequently, the population in 2050 will be roughly:
=> N(2050)=N0*(1/2)*(3.95)=350,000*(0.5)*(3.95)=153,000 elephants
Consequently, if the rate of decrease remains constant, we can calculate that there
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using the fundemental theorem of algebra how many solutions will the function have, f(x)=8x^(3)+216
Pοlynοmial f(x) has exactly 3 cοmplex rοοts
What is Fundamental Theοrem οf algebra?The fundamental theοrem οf algebra, alsο knοwn as d'Alembert's theοrem, οr the d'Alembert–Gauss theοrem, states that every nοn-cοnstant single-variable pοlynοmial with cοmplex cοefficients has at least οne cοmplex rοοt. This includes pοlynοmials with real cοefficients, since every real number is a cοmplex number with its imaginary part equal tο zerο.
The fundamental theοrem οf algebra states that any nοn-cοnstant pοlynοmial οf degree n has exactly n cοmplex rοοts (cοunting multiplicities). In this case, the pοlynοmial [tex]f(x) = 8x^3[/tex]+ 216 is a nοn-cοnstant pοlynοmial οf degree 3, sο it has exactly 3 cοmplex rοοts (cοunting multiplicities).
We can alsο use the factοr theοrem tο cοnfirm that f(x) has exactly 3 rοοts. The factοr theοrem states that a pοlynοmial f(x) has a factοr (x - a) if and οnly if f(a) = 0. In this case, we can factοr οut 8 frοm the pοlynοmial tο get:
[tex]f(x) = 8(x^3 + 27)[/tex]
Setting f(x) = 0, we get:
[tex]8(x^3 + 27) = 0[/tex]
This equatiοn is satisfied if and οnly if [tex]x^3 + 27 = 0.[/tex] We can factοr this equatiοn as fοllοws:
[tex]x^3 + 27 = (x + 3)(x^2 - 3x + 9)[/tex]
The quadratic factοr [tex]x^2 - 3x + 9[/tex] has nο real rοοts, since its discriminant is negative [tex](b^2 - 4ac = (-3)^2 - 4(1)(9) = -27)[/tex]. Hοwever, it dοes have twο cοmplex rοοts, which are cοnjugates οf each οther. Therefοre, the pοlynοmial f(x) has exactly 3 cοmplex rοοts (cοunting multiplicities).
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1000x100/2+8-4=____________
Answer:
[tex]16666\frac{2}{3}[/tex]
Step-by-step explanation:
Solving using PEMDAS
[tex]\frac{1000\times 100}{2+8-4}[/tex]
Since this is a fraction, we can work on the top and the bottom. Lets do the top first. Multiply.
[tex]\frac{100000}{2+8-4}[/tex]
Now we can add two, then subtract.
[tex]\frac{100000}{6}[/tex]
Since this yields an irrational number if we divide, we can simplify this fraction.
[tex]16666\frac{2}{3}[/tex]
Which inequality describes the graph?
Answer:
C
Step-by-step explanation:
dolly buys 12 identical pens for $9.48 how much does each pen cost
Answer:
$0.79
Step-by-step explanation:
$9.48 ÷ 12= 0.79
each pen costs $0.79
The zeros of a function are the values of
for which the function is equal to zero. Enter a number in each blank to make true statements about the function ()=(2−6)(−4)
1) m(x) = 0 when x = 3, and when x = 4.
2) The graph of m intercepts the x-axis at x = 3, and x = 4.
3) The zeros of m are 3 and 4.
1) m(x) = 0 when x = 3, and when x = 4.
To find the zeros of m(x), we set the function equal to zero and solve for x:
m(x) = 0
(2x - 6)(x - 4) = 0
This equation is equal to zero when either 2x - 6 = 0 or x - 4 = 0.
Solving 2x - 6 = 0 gives x = 3, and solving x - 4 = 0 gives x = 4.
2) The graph of m intercepts the x-axis at x = 3, and x = 4.
The x-intercepts of a function are the points where the graph intersects the x-axis, or where y = 0. So, we can find the x-intercepts of m(x) by setting y = m(x) = 0:
m(x) = 0
(2x - 6)(x - 4) = 0
This equation is equal to zero when either 2x - 6 = 0 or x - 4 = 0.
So, the x-intercepts of m(x) are (3, 0) and (4, 0).
3) The zeros of m are 3 and 4.
The zeros of a function are the values of x for which the function is equal to zero. So, the zeros of m(x) are x = 3 and x = 4.
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The given question is incomplete, the complete question is:
The zeros of a function are the values of x for which the function is equal to zero.
Enter a number in each blank to make true statements about the function m(x)=(2x−6)(x−4).
1)m(x) = 0 when x =__, and when x =___
2) the graph of m intercept the x axis at x = __, and x =___ .
3) zeros of m are ___ and ____?
Three times a number lies between negative three and six in digits
Answer:
-3 ≤ 3x ≤ 6
To solve for "x", we can divide each part of the inequality by 3:
-1 ≤ x ≤ 2
Therefore, the number "x" must lie between -1 and 2 in order to satisfy the condition in the sentence.
Step-by-step explanation:
In
△
O
P
Q
,
△OPQ,
Q
O
‾
≅
P
Q
‾
QO
≅
PQ
and
m
∠
Q
=
5
0
∘
.
m∠Q=50
∘
. Find
m
∠
P
.
m∠P
The angle m∠P is equal to 50 degrees.
Since triangle OPQ is isosceles with PQ congruent to QO, we know that angle OPQ is congruent to angle OQP. Let's call this angle x. Then, we can set up an equation based on the fact that the angles in a triangle add up to 180 degrees: x + x + 50 = 180
Simplifying the equation, we get
2x + 50 = 180
Subtracting 50 from both sides, we get
2x = 130
Dividing by 2, we get:
x = 65
Therefore, angle OPQ and angle OQP are both equal to 65 degrees. Since angle OPQ and angle P are supplementary (they add up to 180 degrees), we can find angle P as:
m∠P = 180 - m∠OPQ
=> 180 - 2x
=> 180 - 2(65)
=> 50 degrees.
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i need help i dont know if im right
Since 5/4 is less than 9/4, we know that Alex's rope is shorter than Sam's rope. Since 1 1/4 is greater than 6/5, we know that Brittany's rope is longer than Sam's rope.
What is inequality?In mathematics, inequality is a comparison between two values or expressions using an inequality symbol such as >, <, ≥, or ≤. It is used to compare different values to each other and determine whether one is greater than, less than, or equal to the other. Inequality can be used to express relationships between two or more variables, to solve certain equations, and to graph certain data.
In order to answer these questions, we need to compare the given values. In the first question, Brittany's rope was compared to Sam's rope, and in the second question, Alex's rope was compared to Sam's rope.
For the first question, Sam's rope was 1.5 x 4/5 = 6/5. This value was compared to Brittany's rope which was 1 1/4. Since 1 1/4 is greater than 6/5, we know that Brittany's rope is longer than Sam's rope.
For the second question, Sam's rope was 1.5 x 3/2 = 9/4. This value was compared to Alex's rope which was 5/4. Since 5/4 is less than 9/4, we know that Alex's rope is shorter than Sam's rope.
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Simon and his friends have 27 pieces of candy. They split them up evenly and each person gets 9 pieces. How many people are there? Select the correct equation and solve for p.
Question 6 options:
p/27 = 9; p = 3
9 + p = 27; p = 18
9 = 27 - p; p = 18
27 = 9p; p = 3
Answer:
3 people are getting candy
Step-by-step explanation:
9x3
Two cylinders, A and B, are created. Cylinder B has the same height as Cylinder A. Cylinder B is half the diameter of Cylinder A. Create an expression that presents the volume of cylinder B in terms of the volume of cylinder A,V
What types of symmetry does the shape have?
Answer: 4
Step-by-step explanation:
Given the angles in the figure below, is I1 II I2?
Yes, the line 1 and line 2 are parallel lines as the sum of both given angles is 180°.
Explain about the co-interior angles?Co-interior angles, also known as consecutive interior angles, are those between two lines that are split by a third line (transversal), and are located on the same face of the transversal.
The majority of the time, it comes from the latin word "com-," which often means "along with." Co-interior angles are located on the same face of a transversal as well as between two lines. The significant correlations angles in each diagram are referred to as co-interior angles. Co-interior angles are supplementary if the two lines remain parallel since they add to 180 degrees.
In the given diagram:
= 75 + 105 (co-interior angles)
= 180° (supplementary angles)
So,
line 1 || line 2
Thus, the line 1 and line 2 are parallel lines as the sum of both given angles is 180°.
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Please awnser all 3 like Question 6: awnser, q7: awnser they easy (brainlist)
The amount spent by each friend is 5d + 9.75 = 63.75; d = 10.8 and the mathematics sentence is -10(r + 12.5) = 60.5
How to determine amount spent by each friendIn the question, we have
Total = 63.75
Meal = 9.75 each
Dessert = d each
So, the equation is
5d + 9.75 = 63.75
Evaluate the like terms
5d = 54
Divide by 5
d = 10.8
So, the equation and the result are 5d + 9.75 = 63.75; d = 10.8
How to determine the mathematics sentenceHere, we have
Negative ten times the sum of a number and 12.5 is 60.5.
Let the number be r
So, we have
-10(r + 12.5) = 60.5
How to determine the cost of the jalapeno peppers.Based on the problem statement, we have
8.94 * 3 = 1/3 * (17.95 + j)
So, we have
80.46 = 17.95 + j
Evaluate
j = 62.51
Hence, the cost of the jalapeno peppers is $62.51
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IS IT TRUE ALWAYS SOMETIMES OR NEVER PLEASE HELO ME OR EXPLAIN HOW IK SUPPOSED TO KNOW THE AMSWER
The answer of the question based on the angles statements the answer is
always true based on condition.
What is Range?Range is the difference between highest and lowest values.
Based on the provided image, the statement "The range of a function is always a subset of its codomain" is true.
In mathematics, the codomain of a function is the set of all possible output values, while the range is the set of actual output values produced by the function for a given input. Since the range is a subset of the codomain, the statement is true.
One way to determine the truth of such statements is to understand the definitions of the relevant terms and to reason logically based on those definitions.
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The projected value (in millions of dollars) of a large manufacturing company is modeled by the function V(t) = 230(1. 12)t, where V(t) is the value of the company t years after 2018. What does 230 represent in the function?
In the function [tex]V(t) = 230(1.12)^t[/tex], the number 230 represents the initial or starting value of the company in millions of dollars at the beginning of the time period in question, which is 2018.
The function models the growth of the company's value over time as an exponential function, with a base of 1.12. The exponent t represents the number of years that have passed since 2018, and the resulting value V(t) is the projected value of the company t years after 2018, given the assumed growth rate of 12% per year.
So, at the start of 2018, the company was worth 230 million dollars, according to the model.
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Frank needs to find the area enclosed by the figure. The figure is made by attaching semicircles to each side of a 56-m-by-56-m square. Frank says the area is 1,787. 52m squared. Find the area enclosed by the figure. Use 3. 14 for pi. What error might have made?
The figure's overall area is 8,065.76 square metres (3,136 + 4,929.76 square metres).
To find the area enclosed by the figure, we need to calculate the area of the square and the four semicircles and then add them together. The area of the square is 56 × 56 = 3,136 square meters.
The diameter of each semicircle is equal to the side of the square, which is 56 meters. Therefore, the radius of each semicircle is 28 meters. The area of one semicircle is (1/2) × pi × 28² = 1,232.44 square meters. The area of all four semicircles is 4 × 1,232.44 = 4,929.76 square meters.
Thus, the total area of the figure is 3,136 + 4,929.76 = 8,065.76 square meters.
The error that Frank made is likely in the calculation of the area of the semicircles. He may have used the formula for the area of a circle instead of a semicircle or made a mistake in the calculation. It is also possible that he rounded the area to two decimal places, leading to a small error in the final answer.
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What is (4 - i) + (3 - i) = ?
Answer: 7-2 i
Step-by-step explanation:
Triangle RST was dialated by a scale factor of 3 to create triangle LMN. If TAN R = 5/2.5 find the measurments for LM and MN
The measurements for LM is 30 units and MN is 15 units.
What is the measurement of MN and LM?We know that the scale factor for the dilation is 3, which means that each side of triangle RST was multiplied by 3 to create triangle LMN.
Therefore:
LM = 3(RS)
MN = 3(ST)
To find RS and ST, we can use the fact that TAN R = 5/2.5.
Recall that tangent is the ratio of the opposite side to the adjacent side of a right triangle, so we can set up the following equation:
TAN R = RS/ST = 5/2.5
Cross-multiplying, we get:
RS = 5ST/2.5
Simplifying:
RS = 2ST
Now we can substitute this expression for RS into the equations for LM and MN:
LM = 3(RS) = 3(2ST) = 6ST
MN = 3(ST)
So the measurements for LM and MN are:
LM = 6ST
MN = 3(ST)
From triangle RST,
ST = 5
LM = 6 x 5 = 30 units
MN = 3 x 5 = 15 units
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The wheelchair ramp at the entrance to a store is 12 feet long and rises a total vertical distance of ¾ of a foot. To the nearest degree, what is the angle of inclination of the ramp? Enter the number only.
In response to the question, we may say that As a result, the angle of trigonometry inclination of the ramp is 4 degrees, to the closest degree.
what is trigonometry?The area of mathematics called trigonometry examines how triangle side lengths and angles relate to one another. The subject first came to light in the Hellenistic era, about in the third century BC, as a result of the use of geometry in astronomical investigations. The area of mathematics known as exact techniques deals with several trigonometric functions and possible computations using them. There are six common trigonometric functions in trigonometry. These go by the designations sine, cosine, tangent, cotangent, secant, and cosecant, respectively (csc). Trigonometry is the study of triangle characteristics, particularly those of right triangles. Consequently, studying geometry entails learning about the characteristics of all geometric forms.
The ratio of an angle's opposing side (such as the rise of a ramp) to its adjacent side is known as the tangent (the length of the ramp).
Angle's tangent is equal to its opposite and adjacent sides.
In this instance, the ramp's opposite side represents its 3/4-foot rise, and the ramp's adjacent side is its 12-foot length.
Angle's tangent is equal to 3/4 / 12 (or 0.0625).
We may take the inverse tangent (or arctangent) of this number to determine the angle itself.
Amount = arctan (0.0625)
Calculating the answer, we obtain:
Angle is 3.58 9 degrees.
When we convert this to degrees, we get:
a 4 degree angle
As a result, the angle of inclination of the ramp is 4 degrees, to the closest degree.
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4. Theresa wants new carpeting for her family room. Her family room is a 12 ft by 21 ft rectangle. How much carpeting does she need to buy to cover her entire family room?
Answer:
252 square
Step-by-step explanation:
Let n be a positive integer. If (1+2+3+4+5+6)^2 = 1^3+2^3+. N^3, what is the value of n?
PLEASE HELP :|
The value of n is 3, n is a positive integer
We know that:
1 + 2 + 3 + 4 + 5 + 6 = 21
Therefore:
(1 + 2 + 3 + 4 + 5 + 6)² = 21² = 441
Now, let's look at the sum of cubes:
1³ + 2³ + ... + n³ = (1 + 2 + ... + n)²
We already know that 1 + 2 + ... + 6 = 21, so we can rewrite the equation as:
1³ + 2³ + ... + n³ = (1 + 2 + ... + n)²
1³ + 2³ + ... + n³ = (1 + 2 + 3 + 4 + 5 + 6 + ... + n)²
We want to find the value of n that makes this equation true. We know that the sum of the first n positive integers is:
1 + 2 + 3 + ... + n = n(n+1)/2
So we can rewrite the equation as:
1³ + 2³ + ... + n³ = [n(n+1)/2]²
Now we substitute the value we know for 1 + 2 + 3 + 4 + 5 + 6:
441 = [6(7)/2]²
441 = 21²
So n(n+1)/2 = 7, which means:
n(n+1) = 14
The only positive integer solution for n in this case is 3, because:
n(n+1) = 14
n² + n - 14 = 0
(n-3)(n+4) = 0
The positive integer solution is n = 3, which means:
1³ + 2³ + 3³ = [3(4)/2]² = 36² = 441
So the value of n is 3.
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Prove the identity.
Sec^2 x/2 tan x = csc2x
Note that each Statement must be based on a Rule chosen from the Rule menu. To see a detailed description of a Rule, select the Mor the right of the Rule.
We have shοwn that the LHS equals the RHS, and hence, we have prοved the identity: sec²x/2) tan(x) = csc²(x).
What is trigοnοmetry?Trigοnοmetry is a branch οf mathematics that deals with the study οf relatiοnships between the sides and angles οf triangles. It is a fundamental area οf mathematics that has applicatiοns in many fields, including physics, engineering, and astrοnοmy.
What are the functiοns οf trigοnοmetry?Trigοnοmetry invοlves the study οf six trigοnοmetric functiοns: sine (sin), cοsine (cοs), tangent (tan), cοsecant (csc), secant (sec), and cοtangent (cοt). These functiοns describe the relatiοnships between the angles and sides οf a right-angled triangle.
Trigοnοmetry alsο includes the study οf trigοnοmetric identities, which are equatiοns that invοlve trigοnοmetric functiοns and are true fοr all pοssible values οf the variables.
In the given question,
Starting with the left-hand side (LHS) of the given identity:
sec²(x/2) tan(x)
Using the identity, sec²(x) = 1/cos²(x), we can write:
sec²(x/2) = 1/cos²(x/2)
Substituting this into the LHS:
1/cos²(x/2) * tan(x)
Now, using the identity, tan(x) = sin(x)/cos(x), we can write:
1/cos²(x/2) * sin(x)/cos(x)
Rearranging and simplifying:
sin(x) / cos(x) * 1/cos²(x/2)
Using the identity, csc(x) = 1/sin(x), we can write:
1/sin(x) * 1/cos(x) * 1/cos²(x/2)
Now, using the identity, cos(2x) = 1 - 2sin²(x), we can write:
cos(x) =√(1 - sin²(x/2))
Substituting this into the above equation:
1/sin(x) * 1/√(1 - sin²(x/2)) * 1/cos²(x/2)
Simplifying:
1/sin(x) * 1/√(cos²(x/2)) * 1/cos²(x/2)
Using the identity, csc²(x) = 1/sin²(x) and simplifying:
csc²(x) * cos²(x/2) / cos²(x/2)
The cos²(x/2) terms cancel out, leaving:
csc²(x).
Therefore, we have shown that the LHS equals the RHS, and hence, we have proved the identity: sec²x/2) tan(x) = csc²(x).
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Solve System of Equations from Context (Graphically)Taylor and her children went into a movie theater and she bought $81 worth of bags of popcorn and candies. Each bag of popcorn costs $9 and each candy costs $4.50. She bought 6 more candies than bags of popcorn. Graphically solve a system of equations in order to determine the number of bags of popcorn, x,x, and the number of candies, y,y, that Taylor bought.
Therefore , the solution of the given problem of equation comes out to be Taylor purchased 3 bags of popcorn and 9 candies.
What is equation?The use of the same variable word in mathematical formulas frequently ensures agreement between two assertions. Mathematical equations, also referred to as assertions, are used to demonstrate expression the equality of many academic figures. Instead of dividing 12 into 2 parts in this instance, the normalise technique adds b + 6 to use the sample of y + 6 instead.
Here,
Let's describe our variables first:
x is the quantity of popcorn bags bought, and y is the quantity of sweets bought.
The following system of equations can be constructed using the information provided:
The price of the packages of popcorn and candies is $81, or
=> 9x + 4.5y.
=> y = x + 6 (Taylor purchased 6 more candies than bags of popcorn)
The first step in solving this system of equations numerically is to rewrite the first equation in slope-intercept notation as follows:
=> 4.5y = -9x + 81
=> y = (-2)x + 18
Let's plot the y-intercept at (0,6) and then locate another point by moving up 1 unit and to the right 1 unit to graph the second equation, y = x + 6. This demonstrates our argument (1,7).
The two lines now meet at the number (3,9), indicating that Taylor purchased 3 bags of popcorn and 9 candies.
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a florist is making bunches of flowers for a wedding each bunch contains some carnations, roses and lilies. each bunch is the same. a bunch contains 40 flowers. 60 percent of the flowers are lilies. the ratio of carnations to roses is 3:5. the florist only has 140 carnations. How many bunches of flowers can the florist make?
The florist can make 23 bunches of flowers with the lilies, carnations, and roses they have for the wedding.
What is the ratio?
A ratio is a comparison of two or more quantities that indicates their relative sizes or amounts. It expresses the relationship between two or more numbers or values by dividing one by the other.
Let's start by finding out how many flowers in each bunch are lilies.
Since 60 percent of the flowers are lilies, we can calculate that 0.6 x 40 = 24 of the flowers in each bunch are lilies.
That means the remaining 16 flowers in each bunch are split between carnations and roses.
Let's use the ratio of 3:5 to split these flowers between carnations and roses.
First, we need to find the total number of parts in the ratio:
3 + 5 = 8
That means for every 8 flowers in the bunch, 3 are carnations and 5 are roses.
Now we can set up an equation to find out how many bunches of flowers the florist can make:
140 (number of carnations) ÷ 3 (number of carnations in each 8-flower group) = 46.67
So the florist can make 46 bunches of flowers with the carnations they have.
But we need to make sure that there are enough roses to go with these carnations.
If 3 of every 8 flowers in each bunch are carnations, that means 5 of every 8 flowers are roses.
So for each bunch, there are 5/8 x 16 = 10 roses.
To make 46 bunches of flowers, the florist will need 46 x 16 = 736 flowers in total.
If 24 of every 40 flowers in each bunch are lilies, that means 16 of every 40 flowers are carnations and roses combined.
So for 736 flowers, there will be 368 carnations and roses.
Since each bunch contains 16 flowers, the florist can make 368 ÷ 16 = 23 bunches of flowers with the carnations and roses they have.
Therefore, the florist can make 23 bunches of flowers for the wedding.
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gina prepara un postre para 8 personas usa 1/2 de libra de mantequilla 1/4 de libra de azucar ,una lib a de harina y 3/2 libra de queso cuantas libras de ingredientes necesita si para preparar la receta para 16 personas cuantas libras necesita
Considering a recipe of the dessert for 16 people, the needed amounts of butter, sugar, flour and cheese are given as follows:
Butter: 1 lb.Sugar: 0.5 lb.Flour: 2 lb.Cheese: 3 lb.How to obtain the amounts?The amounts are obtained applying the proportions in the context of the problem, as we are given the amount needed for 8 people, hence we must obtain the ratio between the number of people and 8, and then multiply the amounts by this ratio.
For 8 people, the amounts of the ingredients are given from the problem as follows:
Butter: 0.5 lb.Sugar: 0.25 lb.Flour: 1 lb.Cheese: 1.5 lb.The ratio between 16 people and 8 people is given as follows:
16/8 = 2.
Hence the amount of each ingredient will double, thus the needed amounts are given as follows:
Butter: 1 lb.Sugar: 0.5 lb.Flour: 2 lb.Cheese: 3 lb.TranslationGina is preparing a recipe for 8 people, and the amounts are given as follows:
Butter: 0.5 lb.Sugar: 0.25 lb.Flour: 1 lb.Cheese: 1.5 lb.The problem asks for the necessary amounts for 16 people.
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