The quotient rule in calculus is a method used to find the derivative of any function given in the form of a quotient obtained from the result of the division of two differentiable functions.
That means if we are given a function of the form: f(x) = u(x)/v(x),
we can find the derivative of this function using the quotient rule derivative:
f'(x) = [u(x)/v(x)]' = [v(x) × u'(x) - u(x) × v'(x)]/[v(x)]2
Where:
f(x) = The function of the form u(x)/v(x) for which the derivative is to be calculated.u(x) = A differentiable function that makes numerator of the function f(x).u'(x) = Derivative of function u(x).v(x) = A differentiable function that makes the denominator of the given function f(x).v'(x) = Derivative of the function v(x).To know more about the quotient rule:
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cosec⁴A-cosec ²A=cot ⁴A-cot ²A
from the internet it's different and i did in in this way :
cosec ²A)²-cosec ²A
1+cot ⁴A-2cot²A-1+cot ²A(1-1=0)(2cot ²A-cot ²A=cot ²A)
cot ⁴A-cot ²A
Therefore , the solution of given trigonometric equation has been proved.
The trigonometric equation is what?A trigonometric equation is one that contains one or more ratios of unknown trigonometric angles. The ratios of sine, cosine, tangent, cotangent, secant, and cosecant angles are used to express it. A trigonometric equation is cos2 x + 5 sin x = 0, for instance.
Here,
Given :
Cosec A cosec²A cot A+ cot² A
LHS=(1+cot2A)2- (1 + cot2 A)
=1+2cot2A+cot A-1- cot2 A
= cot2A+ cot1A
= RHS
Hence, Cosec1A- cosec² A = cot ⁴A-cot ²A
Therefore , the solution of given trigonometric equation comes out to be it has been proved
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Find the rule and the graph of the function whose graph can be obtained by performing the translation 3 units right and 4
units up on the parent function f(x)=x²
a. f(x)=(x-5)² +4
8
6-
4
2+
864 -2
-2
L
19
do
+
Mark this and return
+
b. f(x) = (x+3)² + 2
2 4 6
C. f(x)=(x-3)² +4
N
864
d. f(x)=x²-4
2+
-2
4
Top
+
2 4 6
3+
Next
Submit
Answer:
C. f(x) = (x - 3)² + 4
Step-by-step explanation:
Given parent function:
[tex]f(x)=x^2[/tex]
When a graph is translated "a" units right, subtract "a" from the x-value of the function.
Therefore, the translation of the parent function 3 units right is:
[tex]\implies f(x-3)=(x-3)^2[/tex]
When a graph is translated "a" units up, add "a" to the function.
Therefore, the translation of the function 4 units up is:
[tex]\implies f(x-3)+4=(x-3)^2+4[/tex]
Is it possible to draw a triangle has set of sides 9cm 7cm and 17cm give the reason?
It is not possible to construct a triangle whose sides are 9 cm 7 cm and 17 cm because the sum of two sides(9 cm and 7 cm) are smaller than third sides(17 cm).
In the given question, we have to check that it is possible to construct a triangle whose sides are 9 cm 7 cm and 17 cm.
To check whether the given sides can make a triangle or not, we have to check that the sum of two sides always greater than the third side.
To check this we firstly add the 9 and 7
9 + 7 < 17
16 < 17
Now we add 17 and 7
17 + 7 > 9
24 > 9
Now we add 9 and 17
9 + 17 > 7
26 > 7
It is not possible to construct a triangle whose sides are 9 cm 7 cm and 17 cm because the sum of two sides(9 cm and 7 cm) are smaller than third sides(17 cm).
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Help me! Thank you!
Bing chilling
Equation of constant proportionality is y = kx or y = k/x,
she pay for 40 party favours is 3.125
What is Constant of Proportionality?If the ratio of one variable to the other is constant, then the two variables have a proportional relationship, If x and y have a proportional relationship, the constant of proportionality is the ratio of y to x. Sometimes, we also represent it as x is to y.the rate of change also know as the constant of proportionality (k = y/x) which is the constant ratio between two proportional quantities y/x denoted by the symbol k which may be a positive rational number. The x value is directly proportional to the y value such as in the equation y = kx.We use constant of proportionality in mathematics to determine the nature of proportionality, whether it is direct proportion or indirect proportion. The constant of proportionality helps in solving the equations involving ratios and proportions.To learn more about Constant of Proportionality refers to:
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Convert this equation into standard form
The standard form of the given linear equation is; -x + 3y = 15
How to write the Linear Equation in standard form?The standard form of a linear equation is expressed in the form;
Ax + By = C
Now, we are given the equation as;
y - 4 = ¹/₃(x + 3)
Multiply through by 3 using multiplication property of equality to get;
3y - 12 = x + 3
Add 12 to both sides to get;
3y = x + 15
Subtract x from both sides to get;
-x + 3y = 15
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Which of the following are factors of 56? A.4 B.6 C.7 D.9 E.10
Answer:
Step-by-step explanation:
A and C will both work. 4 goes into 56 evenly and so does 7.
What are the 4 steps to solving equations?
An equation can be solved in four steps.
1. Find the equation's terms and coefficients.
2. To isolate the variable, using inverse techniques.
3. Re-enter the value into the first equation to verify your conclusion.
4. Consider the issue in light of the remedy.
Finding an equation's terms and coefficients is the first step in solving it. Any constants, variables, and coefficients must be located, and their relationships must be understood.Once the terms are known, it is feasible to isolate the variable and perform inverse operations to solve for the unknown, such as addition, subtraction, multiplication, and division.
It is crucial to check the solution after solving the issue by inputting the original formula once more. The answer is accurate if the equation still holds true. Understanding the answer requires consideration of the problem's context. This necessitates figuring out the response's applicability and how to apply it to the bigger issue.
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stressed bc test due tonight help plese ty
For 16-19, graph each equation or inequality.
16. 2y – 4x < 8
17. -4y > -x + 12
18. | x + 3 |= y
19. |2x – 6| + 2 = y
The solution is the set of all points (x,y) that satisfy the inequalities:
y < 2x + 4, y < x/4 - 3, x + 3 = y, and 2x - 6 + 2 = y.
What is the graph of a function?
The graph of a function is a visual representation of the relationship between the input values (x-coordinates) and the output values (y-coordinates) of the function. In other words, it is a visual representation of the function's rule.
The graph of a function can take many different forms, depending on the function's rule. Linear functions, for example, will produce a straight line on the graph.
i) 2y - 4x < 8 can be solved by adding 4x to both sides and then dividing by 2, giving:
y < 2x + 4.
ii) -4y > -x + 12 can be solved by adding x to both sides and then dividing by -4, giving: y < x/4 - 3.
iii) | x + 3 | = y, you can split this into two cases:
i. x + 3 = y
ii. -(x + 3) = y
iv) |2x – 6| + 2 = y, you can split this into two cases:
i. 2x - 6 + 2 = y
ii. -(2x - 6) + 2 = y
We can use these equations to plot the graph on a coordinate plane.
Hence, the solution is the set of all points (x,y) that satisfy the inequalities:
y < 2x + 4, y < x/4 - 3, x + 3 = y, and 2x - 6 + 2 = y.
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What is an equation of the line, in point-slope form, that passes through the given point and has the given slope? point: (9,3); slope: 4/9
A. y-9=4/9(x+3)
B. y-9=4/9(x-3)
C. y+3=4/9(x-9)
D. y-3=4/9(x-9)
The equation of the line passing through point (9, 3) and having a slope of 4/9 is y - 3 = (4/9)(x - 9)
What is an equation?An equation shows how two or more numbers and variables are related to each other.
The standard linear equation is:
y = mx + b
Where m is the rate of change and b is the y intercept
The equation of a line passing through the point (x₁, y₁) and having a slope of m is given as:
y - y₁ = m(x - x₁)
Given that a line passes through point (9, 3) and have a slope of 4/9, the equation is:
y - 3 = (4/9)(x - 9)
The equation of the line is y - 3 = (4/9)(x - 9)
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Kevin drew a scale drawing of a house. The scale he used was 1 millimeter : 4 meters. If the actual length of the living room is 8 meters, how long is the living room in the drawing?
If the actual length of the living room is 8 meters, according to the scale drawing living room in the sketching should be 2 millimeters
What is the scale drawing?Scale drawings are images that depict objects at a scale different from their actual size. Depending on the size of the object being represented and how the sketching will be used, they can depict objects at either a bigger or smaller scale than full size. The scale gives the proportion of a distance at full size to the distance that, at the scale in use, would have the same length.
Scale drawings are used to illustrate objects when sketching them at their true scale would not be practical or beneficial. This could be because depicting the object as its whole would be difficult or difficult to fit on a single sheet of paper.
since 4meter = 1mm
then 8meter = 2mm
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Is y² 5y 10 a quadratic equation justify?
Yes, y² = 5y - 10 is a quadratic equation.
In algebra, a quadratic equation is that which can be rearranged in standard form as
ax² + bx + c = 0,
where x is an unknown value, and a, b, and c are known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear). Quadratic equations are the polynomial equations of degree 2 in one variable of type f(x) = ax² + bx + c = 0 where a, b, c, ∈ R and a ≠ 0.
We have the equation
y² = 5y - 10
y² - 5y + 10 = 0
The highest degree of this equation is 2, therefore it is a quadratic equation.
-- The given question is incorrect, the correct form of equation is
"Is y² = 5y - 10 a quadratic equation justify?" --
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Is the absolute value of a complex number real?
The absolute value of a complex number is real.
The absolute value of a complex number is also known as its magnitude. It is a non-negative scalar quantity that represents the distance of the complex number from the origin in the complex plane.
The absolute value of a complex number is represented by the symbol "|z|" where z is the complex number.
The absolute value of a complex number is always a real number. For example, the absolute value of the complex number 4 + 3i is 5 (the distance of this point from the origin), which is a real number.
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Which of the following equations have an -intercept of (-2, 0) and y-intercept of (0, 3)?
Select all that apply.
y=3/2x+3
y=3x-2
2y-3x=6
y=-2x+3
y-1=-2(x+3)
Answer:
The first and third choices
[tex]y = \dfrac{3}{2}x + 3[/tex]
and
[tex]2y-3x=6[/tex]
Step-by-step explanation:
(I cannot properly see the leading negative signs in choices 4 and 5, I copied/pasted your question choices from Brainly post and zoomed)
First understand what x and y intercepts are
The y-intercept is the value of y where the graph of the equation crosses the y-axis, It is the value of y when x = 0
The x-intercept is the value of x where the graph of the equation crosses the x-axis, It is the value of x when y = 0
We are given that the x-intercept is (-2, 0) and y = (0,3)
We can use a process of elimination to get rid of obviously incorrect choices and then work on the possible correct ones depending on whether we come up with the correct values of y first and eliminating those that do not match x =0, y =3 or (0,3) and then see if the x-intercept is (-2,0)
First choice let's find the y-intercept for the first choiceHow do you enlarge a triangle with scale factor 1 2?
Every coordinate point of the original triangle is multiplied by the scale factor 1/2 if the scale factor is 1/2. multiplying by half after a scale factor of two.
What is Scale Factor?The scale factor of a shape refers to the amount by which it is increased or shrunk. It is applied when a 2D shape, such as a circle, triangle, square, or rectangle, needs to be made larger.
K is the scaling factor for x in the equation y = Kx. This expression can also be visualised in terms of proportionality:
y ∝ x
K can therefore be regarded as a proportionality constant in this context.
Basic Proportionality Theorem also aids in a better understanding of the scale factor.
Shapes in various dimensions can be scaled using the Scale Factor. In geometry, we study many geometrical forms that exist in both two- and three-dimensions. The scale factor is a way to compare figures with similar appearances but differing scales or measurements. Consider two circles that resemble one another but may have different radii.
The scaling factor indicates how much a figure has increased or decreased from its initial value. With the use of the scale factor, any original shape can be drawn in an enlarged or shrunk form.
Formula of Scale Factor
Dimensions of Original Shape x scale Factor = Dimension of new shape
Scale factor = Dimension of New Shape/Dimension of Original Shape
What is Dilation?The process of dilation entails scaling or changing a thing. It is a transformation that uses the specified scale factor to make the objects smaller or larger. The initial figure is referred to as the pre-image, and the new figure that results from dilatation is known as the image. Dilation comes in two varieties:
Increase in an object's size is referred to as expansion.Contraction is the reduction of an object's size.How to enlarge a triangle with scale factor ?The scale factor (or ratio) and the centre of the dilation are included in a dilation's description.A fixed point in the plane serves as the centre of dilation.The image is an enlargement if the scale factor is greater than 1. (a stretch). The image is a decrease if the scale factor is between 0 and 1. (a shrink). The figure and the image are consistent if the scale factor is 1.To know more about Scale Factor,Refer to:
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Every coordinate point of the original triangle is multiplied by the scale factor 1/2 if the scale factor is 1/2. multiplying by half after a scale factor of two.
What is Scale Factor?The scale factor of a shape refers to the amount by which it is increased or shrunk. It is applied when a 2D shape, such as a circle, triangle, square, or rectangle, needs to be made larger.K is the scaling factor for x in the equation y = Kx. This expression can also be visualised in terms of proportionality:y ∝ xK can therefore be regarded as a proportionality constant in this context.Basic Proportionality Theorem also aids in a better understanding of the scale factor.Shapes in various dimensions can be scaled using the Scale Factor. In geometry, we study many geometrical forms that exist in both two- and three-dimensions. The scale factor is a way to compare figures with similar appearances but differing scales or measurements. Consider two circles that resemble one another but may have different radii.The scaling factor indicates how much a figure has increased or decreased from its initial value. With the use of the scale factor, any original shape can be drawn in an enlarged or shrunk form.Formula of Scale Factor
Dimensions of Original Shape x scale Factor = Dimension of new shape
Scale factor = Dimension of New Shape/Dimension of Original Shape
What is Dilation?The process of dilation entails scaling or changing a thing. It is a transformation that uses the specified scale factor to make the objects smaller or larger. The initial figure is referred to as the pre-image, and the new figure that results from dilatation is known as the image. Dilation comes in two varieties:Increase in an object's size is referred to as expansion.Contraction is the reduction of an object's size.How to enlarge a triangle with scale factor ?The scale factor (or ratio) and the center of the dilation are included in a dilation's description.A fixed point in the plane serves as the center of dilation.The image is an enlargement if the scale factor is greater than 1. (a stretch).The image is a decrease if the scale factor is between 0 and 1. (a shrink).The figure and the image are consistent if the scale factor is 1.To know more about Scale Factor visit:
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Let I be the flux of G = (9e^y, 3x^2 ex^3 , 0) through the upper hemisphere S of the unit sphere.
(a) Find a vector field A of the form (0, 0,...) such that curl(A) = G.
(b) Calculate the circulation of A around 88.
(c) Compute the flux of G through S (a)
A= _____ +(C1,C2,C3)
(b) integrate A .ds =
(c) I =
Answer:
check Here.
Step-by-step explanation:
(a) A can be found by solving the equation curl(A) = G, which means that the curl of A in the x, y, and z directions must equal the corresponding components of G. To find A, we can use the vector identity curl(A) = del x A - del y A + del z A.
From this, we get:
del y A = 9e^y
del x A = -3x^2 ex^3
del z A = 0
So A = (f(z), g(x, y), h(x, y, z)) where f, g, h are arbitrary differentiable functions.
(b) Circulation of A around C is given by the line integral of A . ds, where C is a closed curve and ds is an infinitesimal element of C. Since we are given a specific curve 88, we need to know the parametric representation of the curve to calculate the circulation.
(c) The flux of G through S is given by the surface integral of G . dS, where dS is an infinitesimal element of the surface S. Since we are given the upper hemisphere of the unit sphere as S, we can use spherical coordinates to parametrize the surface and then use the divergence theorem to calculate the flux.
Note that the specific values of A, the circulation, and the flux are dependent on the choice of f, g, h and the representation of 88, and dS.
Gwen can a photo that i 4inche wide by 6 inche long into her computer. If he cale the length down to 3inche how wide hould the imilar photo be?
The width of a similar photo will be 2 inches.
As per the given problem,
Width of original photo = 4 inches
Height of original photo = 6 inches
After scaling the photo,
Height of new photo = 3 inches.
Let’s understand about scale factor.
An amount that multiplies (or "scales") a quantity is called a scale factor.
Scale factor = height of the sized image/size of the original image.
Scale factor = 3/6 = 1/2.
Therefore, the scaled image's width = scale factor *the size of the original image.
The width of the scaled photo = (½)*4 = 2.
Hence, the width of a comparable image measures 2 inches.
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what is equivlent to 4/9x7
Answer:
Step-by-step explanation:
The expressions that are equivalent to 4/9 x 7 are 28/9, 3¹/₉ and 3.11
As no options are included, I will include the expression that is equivalent to 4/9 x 7 in both fractions and decimal.
In fractions:
= 4/9 x 7
= 28/9
or,
= 3¹/₉
In decimals:
= 4/9 x 7
= 28/9
= 28 ÷ 9
= 3.11
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ninas garden is 4 1/5 meters long 3/10 meters wide . what is the area of ninas garden? please help asap
Based on the information provided, the area in Nina's garden is 0.26 square meters.
What is the area?The word area refers to the space occupied by a surface, in this case, the space of the garden. In a rectangle, this area can be calculated with the following formula, although for other shapes the formula might be different:
area = length x width
What is the area of the garden?To make this process easier, let's convert the fractions into regular numbers
4 1/5 = 4.2 (1/5 = 0.2)
3/10 = 0.3
4.2 x 0.3 = 1.26 square meters
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A 50centimeter picece of wire is bent into a cricle. What is the area of this cricle?
The area of the circle is 1962.5cm²
What is area of a circle?The area is the region bounded by the shape of an object. The space covered by the figure or any two-dimensional geometric shape, in a plane, is the area of the shape.
Area of a circle is the region occupied by the circle in a two-dimensional plane. It can be determined easily using a formula, A = πr², (Pi r-squared) where r is the radius of the circle.
The length of the wire will be the diameter of the circle. Therefore the radius of the circle = 50/2 = 25 cm
Area of a circle is given as = πr²
A = 3.14×25×25
A= 1962.5 cm²
Therefore the area of the circle when a length 50cm of wire is bent into circle is 1962.5cm²
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Which quadrant the point p(2,3) and q(-3,2) lie?
The given points lies in the following quadrant :
Point P ( 2,3 ) lies in first quadrant and point Q( -3,2 ) lies in the second quadrant.
As given in the question,
Given points are as follow:
Coordinates of point P are ( 2, 3)
Coordinates of point Q are ( -3, 2)
Sign of coordinates in each quadrant of the coordinate plane:
Let ( x, y) be the coordinates.
First quadrant : ( x , y) both are positive
Second quadrant : ( -x , y)
Third quadrant : ( -x , -y )
Fourth quadrant : ( x , -y )
Point P( 2,3) represent both are positive:
Point P lies in the first quadrant.
Point Q(-3,2) represent (-x ,y).
Point Q lies in the second quadrant.
Therefore, point P (2,3) lies in the first quadrant and point Q(-3,2) lies in the second quadrant.
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The corresponding angles in these polygons are congruent.
3/1107
Long division please show work! Tysm
What is table of numbers?
A table of numbers is a collection of numbers arranged in a specific format, usually in rows and columns.
Write the uses of table of numbers.The numbers in the table can represent various types of data, such as measurements, counts, or statistical values. Tables of numbers can be used to organize, compare, and analyze data in a clear and easy-to-read format. They are often used in fields such as math, statistics, and data analysis. Tables can be created using spreadsheet software, or they can be created manually using a pen and paper. The table can be used to represent a function, a data set, a matrix or a multiplication table, etc
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Complete question is: What is table of numbers?
Solve this problem using 45-45-90
The right angled triangle's side measures 19, its hypotenuse is approximately 8100.25.
what is triangle ?Since a triangle has three sides and three vertices, it is a polygon. It is a fundamental geometric shape. Triangle ABC is the moniker given to a triangle that has vertices A, B, and C. When the three points are not collinear, a singular plane and triangle are found in Euclidean geometry. A triangle is a polygon if it has three sides and three corners.
given
Let's say we don't know the triangle's hypotenuse and that angle A is approximately 33.56.
Cos and Cos-1 are available. Make sure it is set to degrees.
Cos(33.56)=6750/h
h=8100.25
The right angled triangle's side measures 19, its hypotenuse is approximately 8100.25.
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the perimiter of a rectangle is 30 inches. if its length is three times its width, find the dimensions
The answer is l = 3.75 and l = 11.25.
Given the length l and width w of a rectangle, its perimeter is:
P=2(l+w)
This formula can be used to form an equation as we already know the value of P.
So, Let the width be w. The length will be 3w. As the perimeter is 30 inches, the width can be found as follows.
2(w+3w)=30
4w=15
w=3.75
Thus, the width is 3.75 inches. The length will be 3*3.75=11.25 inches.
Steps:
Step 1: Let the width of the rectangle be w. Step 2: Since the length is 3 times the width, the length of the rectangle is 3w. Step 3: The perimeter of the rectangle is 30 inches. Step 4: We can write the formula for the perimeter of the rectangle as 2w + 2(3w) = 30. Step 5: Solve for w, then we get w = 3.75Step 6: therefore l = 3w = 3.75 * 3 = 11.25.To know about the Parameters of Rectangle:
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please help its for a test and I need these done so I can be free. its due tonight and I did this all day
For 16-19, graph each equation or inequality.
16. 2y – 4x < 8
17. -4y > -x + 12
18. | x + 3 |= y
19. |2x – 6| + 2 = y
An inequality is defined as an unfair comparison between two numbers or other mathematical expressions. When comparing the sizes of two numbers on a number line, it is most commonly used. The graph for each inequality or equation is included in the attachment.
What is graph?A graph is a visual representation of statistical data or a functional relationship between variables. Graphs are useful for forecasting because they can highlight broad trends in the quantitative behaviour of data. However, as mere approximations, they can be inaccurate and occasionally deceptive.
Most graphs have two axes: one for a set of independent variables and one for a set of dependent variables. The most common type of graph is a broken-line graph, with time serving as the independent variable.
16. 2y – 4x < 8
= 2y < 4x + 8
= y < 2x + 4
17. -4y > -x + 12
= y > x/4 + -3
18. | x + 3 |= y
y = | x + 3 |
19. |2x – 6| + 2 = y
y = |2x – 6| + 2
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struggling please help I'll mark brainliest
What is the solution to the inequality |2x + 3| < 7?
4 < x < 10
–5 < x < 2
x < 4 or x > 10
x < –5 or x > 2
Answer:
its the second one: -5<x<2
Step-by-step explanation:
I did out the math
The length of a rectangle is 12 inches the width is 2 inches
Answer:
The area is 24 inches, the perimeter is 30 inches.
Step-by-step explanation:
Two angles are complementary the difference between four times the measure of the smaller angle and the measure of the larger angle is 10 what is the measure of the larger angle
The measure of the larger angle is obtained as follows:
70º
What are complementary angles?Two angles are defined as complementary is the sum of their measures is of 90º.
The variables that represent each angle in this problem are given as follows:
x and y.
They are complementary angles, hence:
x + y = 90.
The difference between four times the measure of the smaller angle and the measure of the larger angle is 10, hence:
4y - x = 10.
x = 4y - 10.
Hence the value of y is obtained as follows:
4y - 10 + y = 90
5y = 100
y = 100/5
y = 20º.
Hence the measure of the larger angle is given as follows:
x + 20 = 90
x = 70º.
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Let g(x) = 2x and h(x) = x^2 + 4. Evaluate (h ∘ g)(−5).
Answer:
(h ￮ g)(- 5) = 104
Step-by-step explanation:
evaluate g(- 5) , then substitute the value obtained into h(x)
g(- 5) = 2(- 5) = - 10 , then
h(- 10) = (- 10)² + 4 = 100 + 4 = 104