Answer:
17,500
Step-by-step explanation:
Write in Slope-Intercept form:
Slope = -2
Y-Intercept = ( -5 , -4 )
Step-by-step explanation:
The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
Given the slope m = -2 and the y-intercept (a, b) = (-5, -4), we can substitute these values into the equation to get:
y = mx + b
y = -2x + (-4)
y = -2x - 4
Therefore, the equation in slope-intercept form is y = -2x - 4.
x 2 −51=14xx, squared, minus, 51, equals, 14, x 1) Rewrite the equation by completing the square.
Answer:
(x -7)² = 100
Step-by-step explanation:
You want to rewrite the equation x² -51 = 14x by completing the square.
Complete squareWhen we're finished, the equation will be of the form ...
(x +a)² = b
Expanding this, we have ...
x² +2ax +a² = b
That is, the value of 'a' is half the coefficient of x when the equation is written with the x-terms together.
RearrangementAdding 51 -14x to both sides of the given equation, we get ...
x² -51 = 14x
x² -14x = 51
Now, we can see that a=-14/2 = -7. We can add a² = (-7)² = 49 to both sides to complete the square:
x² -14x +49 = 51 +49
(x -7)² = 100 . . . . . . . rewritten equation
You can compare quantities more easily using properties of exponents. For example, you can use exponent properties to compare the weights of 3,124-lb mother hippopotamus and there 125-lb baby. When else might you want to use exponent properties
Exponential property can be used in various field or place like growing or decline of population, bacteria growth/decay , compound interest any many more.
What is Exponential property?Exponential property states that when we multiply two power with same base we add the exponents.
[tex]a^{m} *b^{n } = (a*b)^{m + n}[/tex]
When we divide two power with same base we subtract the exponents.
[tex]a^{m} \div b^{n} = (a \div b)^{m-n}[/tex]
When have to find the power of power we multiply the exponents.
[tex](a^{m} )^{n} = a^{m*n}[/tex]
Three most important use of exponential function is to calculate carbon dating of any soil, fossil foil etc, to calculate population growth , interest earn on an investment etc.
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1.1.1 To determine how much grass seeds they will need, the caretaker measure the length and breadth of the area of the sports field. He measured the length to be 162m and the breadth 160m. Calculate the area where the grass will be planted. You may use the formula: Area of rectangle = length x breadth (2)
Answer:
the area where the grass will be planted is 25,920 square meters.
Step-by-step explanation:
To calculate the area where the grass will be planted, we need to multiply the length and breadth of the sports field.
Area = length x breadth
Area = 162m x 160m
Area = 25,920 square meters
Therefore, the area where the grass will be planted is 25,920 square meters.
What are some mistakes when solving a 2 step equation with a single variable
Answer:Not dividing the entire side of the equation.
Not distributing properly using F.O.I.L.
Confusing the rules for adding and subtracting negative numbers with the rules for multiplying and dividing negative numbers.
Multiplying exponents with the base.
Step-by-step explanation:
5. A rock is thrown directly upward with an initial velocity of 79 feet per second from a cliff 50 feet above a beach. The height of the rock above the beach (h) after t seconds is given by the equation h = -16t² + 79t + 50. The graph below shows the rock's height as a function of time.
The rock will be 125 feet above the beach at: The values of t are 3.911 or 1.28 and (b) The minimum height is h= 346 feet
How to find velocity?Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time
The given parameters are
h = -16t² + 79t + 50
In the equation put h = 125 to find t
125 = -16t² +79t+50
Rearrange the equation to have
16t²-79t -50 +125 = 0
this is also written as 16t² -79t + 75 =0
Solving for t we have
[-b±√b²-4ac]/2z
[79 ±√79²-4*16*75]/2*16
[(79 ±√6241-4800)] / 32
Simplify further to get
[79 ±√1441] / 32
(79 ±38) / 32
117/32 or 41/32
The values of t are 3.911 or 1.28
The minimum height of the rock at t = 1.28 is
h = -16(3.28)² + 79( 3.28)+ 50
h = -16(10.7584) + 79(3.28) +50
h = - 172.1344 +259.12 + 50
h= 346 feet
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Solve for x.
0≤3x-6≤18
A 0≤x≤8
B 2≤x≤8
C x≤0 orx≥8
D x≤2 or x ≥8
Answer:
B) 2≤x≤8.
Step-by-step explanation:
Add 6 to all sides of the inequality.
0+6≤3x-6+6≤18+6
6≤3x≤24
Divide all sides by 3.
2≤x≤8
The correct answer is B) 2≤x≤8.
Write a polynomial function of the least degree with integral coefficients that have the given zeros.1+3i,-2i
The polynomial function of the least degree with integral coefficients that has the zeros [tex]1 + 3i[/tex], [tex]-2i[/tex] is f(x) = x⁴ - 2x³ + 14x² - 8x + 40.
Writing a polynomial function of the least degreeFrom the question, we are to write a polynomial function of the least degree with integral coefficients that have the given zeros.
The given zeros are [tex]1+3i[/tex],[tex]-2i[/tex].
If [tex]1 + 3i[/tex] and [tex]-2i[/tex]are zeros of a polynomial function with integral coefficients, then their conjugates [tex]1 - 3i[/tex] and [tex]2i[/tex] are also zeros of the function.
To find the polynomial function, we can use the fact that if r is a zero of a polynomial function, then (x - r) is a factor of the function. Thus, we can start by writing out the factors corresponding to each of the zeros:
[tex](x - (1 + 3i))(x - (1 - 3i))(x - (-2i))(x - 2i)[/tex]
Next, we can simplify these factors by multiplying them out:
[tex][(x - 1) - 3i][(x - 1) + 3i](x + 2i)(x - 2i)\\= [(x - 1)^2 - (3i)^2](x^2 - (2i)^2)[/tex]
= [(x - 1)² + 9](x² + 4)
Expanding the terms, we get:
(x² - 2x + 10)(x² + 4)
Multiplying out the factors, we obtain:
x⁴ - 2x³ + 10x² + 4x² - 8x + 40
Simplifying this expression, we get:
x⁴ - 2x³ + 14x² - 8x + 40
Hence, the polynomial function is f(x) = x⁴ - 2x³ + 14x² - 8x + 40
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Write an inequality to describe each situation. a. The minimum age for voting in the United States is 18 years old. Let a represent a voter's age. b. A theater seats up to 275 people. Let p represent the number of people attending a performance in the theater.
According to this inequality, the number of persons attending a theatrical inequality play, denoted by p, must be less than or equal to 275 in order for everyone to have a seat.
What is inequality?In mathematics, an inequality is a non-equal connection between two expressions or values. As a result, imbalance leads to inequity. In mathematics, an inequality connects two values that are not equal. Inequality is not the same as equality. When two values are not equal, the not equal symbol is typically used (). Various disparities, no matter how little or huge, are utilised to contrast values. We resuming our current status quo. Yet a variety of things lead to inequality: Negative values on both sides are split or added.
a. The disparity in representing the voting age in the United States is as follows:
a ≥ 18
In order to be eligible to vote, a voter's age, denoted by a, must be more than or equal to 18 years old.
b. The inequality used to depict a theater's seating capacity is:
p ≤ 275
According to this inequality, the number of persons attending a theatrical play, denoted by p, must be less than or equal to 275 in order for everyone to have a seat.
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As economy of coal, electric, and steel industries. For each $1.00 of output, the coal industry needs $0.02 worth of coal, $0.30 worth of electricity, and 0.30 worth of steel, the electric industry needs $0.04 worth of coal, $0.04 worth of electricity, and 0.02 worth worth of steel, and the steel industry needs $0.10 worth of coal and $0.04 worth of steel. The sales demand is estimated to be $1 billion for coal, $1 billion for electricity, and $4 billion for steel. Suppose that the demand for electricity triples and demand for coal doubles, whereas the demand for for steel increases by only 50%. At what levels should the various industries produce in order to satisfy the new demand.
Answer:
Step-by-step explanation:
Let’s denote the production levels of coal, electricity and steel as x, y and z respectively. We can set up a system of equations to represent the inter-industry demand for each industry’s output.
For coal: 0.02x + 0.04y + 0.10z = x For electricity: 0.30x + 0.04y = y For steel: 0.30x + 0.02y + 0.04z = z
Solving this system of equations gives us x = (50/3)y and z = (25/2)y.
The new sales demand for coal is $2 billion (double the original), for electricity is $3 billion (triple the original) and for steel is $6 billion (an increase of 50%). Substituting these values into our equations gives us:
(50/3)y = $2 billion y = $3 billion (25/2)y = $6 billion
Solving these equations gives us y = $3 billion, x = $5 billion and z = $18.75 billion.
So to satisfy the new demand, the coal industry should produce at a level of $5 billion, the electric industry should produce at a level of $3 billion and the steel industry should produce at a level of $18.75 billion.
Which of the following comparisons is false? 5 degrees centigrade warmer than 5 degrees farenheight or 15 degrees centigrade is cooler than 60 degrees farenheight or 30 degrees centigrade centigrade is warmer than 90 degrees farenheight or 35 degrees centigrade cooler than 100 degrees farenheight ?
The false comparison is: 35 degrees centigrade cooler than 100 degrees Fahrenheit.
What is an expression?In mathematics and computer programming, an expression is a combination of one or more values, variables, operators, and functions that are evaluated to produce a result.
According to the given information:The comparison "35 degrees centigrade cooler than 100 degrees Fahrenheit" is false because 35 degrees centigrade is equivalent to 95 degrees Fahrenheit, which is actually warmer than 100 degrees Fahrenheit. Therefore, it is incorrect to say that 35 degrees centigrade is cooler than 100 degrees Fahrenheit.
Therefore, the false comparison is: 35 degrees centigrade cooler than 100 degrees Fahrenheit.
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What is the approximate value of this logarithmic expression?
The approximate value of the given logarithmic expression as required to be determined in the task content is; 1.528.
What is the value of the logarithmic expression as required?It follows from the task content; it is required that the value of the logarithmic expression is to be determined.
Therefore, we have;
let the result of the expression be x; so that we have;
log₈ (24) = x
x log (8) = log (24)
x = log (24) - log (8)
x = 1.528
Ultimately, the approximate value of the logarithmic expression in discuss is; 1.528.
Complete question: The correct expression is; log₈ (24) .
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P12 000 is deposited in an account earning 4% interest per year. What is the amount
after 15 years
Answer:
After 15 years, the amount in the account will be $21.611.32, assuming the interest is compounded annually.
Step-by-step explanation:
To calculate the amount after 15 years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where A is the amount, P is the principal (initial amount), r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time (in years).
In this case, P = $12,000, r = 0.04 (4% expressed as a decimal), n = 1 (compounded annually), and t = 15 years.
Plugging in the values, we get:
A = $12,000(1 + 0.04/1)^(1*15)
A = $12,000(1.04)^15
A = $12,000(1.801)
A = $21,611.32
Therefore, after 15 years, the amount in the account will be $21.611.32, assuming the interest is compounded annually.
Someone help I’ll mark BRAINLIEST
Answer:
1. c
2. b
3. B
Step-by-step explanation:
1. the hypotenuse is the longest side of the triangle.
2. think of it as the leg that helps form angle A (or forms right angle) that is NOT the hypotenuse
3. if "c" is the hypotenuse,, & b is the adjacent leg,, the opposite angle is across from the given angle which in this case is A
Someone help me with this question please. I attached the screenshot. Thanks.
I forgot where to start in solving this equation
Please help i’m struggling
The vertices will be D = (-3,2) E=(1,2) F=(-2,0).
What is dilatiοn?resizing an οbject is accοmplished thrοugh a change called dilatiοn. The οbjects can be enlarged οr shrunk via dilatiοn. A shape identical tο the sοurce image is created by this transfοrmatiοn. The size οf the fοrm dοes, hοwever, differ. A dilatatiοn οught tο either extend οr cοntract the οriginal fοrm. The scale factοr is a phrase used tο describe this transitiοn.
The scale factοr is defined as the difference in size between the new and οld images. An established lοcatiοn in the plane is the center οf dilatatiοn. The dilatiοn transfοrmatiοn is determined by the scale factοr and the centre οf dilatiοn.
Here the scale factοr is 1/2
Sο the vertices will be D = (-3,2)
E=(1,2)
F=(-2,0)
Hence the vertices will be D = (-3,2) E=(1,2) F=(-2,0).
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can someone help me with this
please show work
Answer:
1) S=31.68 yd A=174.23 [tex]yd^{2}[/tex]
2) S=36.65 in A=256.56 [tex]in^{2}[/tex]
3) S=8.9 ft A=13.35 [tex]ft^{2}[/tex]
4)S=17.28 in. A=51.84 [tex]in^{2}[/tex]
5)S=25.31 yd. A=126.54 [tex]yd^{2}[/tex]
6)S=3.4ft A=5.11 [tex]ft^{2}[/tex]
Step-by-step explanation:
Notes:
360 was divided by 1/2, making the denominator 180. This is why there is a 1/2 at the front.
You also do not need to simplify if you're using a calculator.
A=[tex]\frac{1}{2}[/tex]([tex]r^{2}[/tex]) θ
- r is the radius
- θ is [tex]\frac{angle\pi }{180}[/tex]
S= r θ
-r is the radius
- θ is [tex]\frac{angle\pi }{180}[/tex]
1)
S=r θ
S=11([tex]\frac{165\pi }{180}[/tex])........................................plug in values
S=11( [tex]\frac{11\pi }{12}[/tex]).........................................simplify
S=31.68 yd....................................solve and round
A=[tex]\frac{1}{2}[/tex]([tex]r^{2}[/tex]) θ
A=[tex]\frac{1}{2}[/tex]([tex]11^{2}[/tex]) [tex]\frac{165\pi }{180}[/tex]....................................plug in values
A= =[tex]\frac{1}{2}[/tex]([tex]121[/tex]) [tex]\frac{11\pi }{12}[/tex]....................................simplify
A=174.23 [tex]yd^{2}[/tex].....................................solve and round.
2)
S=r θ
S=14([tex]\frac{150\pi }{180}[/tex])...........................................plug in values
S= =14( [tex]\frac{5\pi }{6}[/tex]).........................................simplify
S=36.65 in........................................solve and round
A=[tex]\frac{1}{2}[/tex]([tex]r^{2}[/tex]) θ
A=[tex]\frac{1}{2}[/tex]([tex]14^{2}[/tex]) [tex]\frac{150\pi }{180}[/tex]......................................plug in values
A= =[tex]\frac{1}{2}[/tex]([tex]196[/tex]) [tex]\frac{5\pi }{6}[/tex]......................................simplify
A=256.56 [tex]in^{2}[/tex]....................................solve and round.
3)
S=r θ
S=3([tex]\frac{170\pi }{180}[/tex]).........................................plug in values
S= 3( [tex]\frac{17\pi }{18}[/tex]).......................................simplify
S=8.9 ft..........................................solve and round
A=[tex]\frac{1}{2}[/tex]([tex]r^{2}[/tex]) θ
A=[tex]\frac{1}{2}[/tex]([tex]3^{2}[/tex]) [tex]\frac{170\pi }{180}[/tex]........................................plug in values
A= =[tex]\frac{1}{2}[/tex]([tex]9[/tex]) [tex]\frac{17\pi }{18}[/tex].......................................simplify
A=13.35 [tex]ft^{2}[/tex]........................................solve and round.
4)
S=r θ
S=6([tex]\frac{165\pi }{180}[/tex]).......................................plug in values
S=6( [tex]\frac{11\pi }{12}[/tex]).......................................simplify
S=17.28 in....................................solve and round
A=[tex]\frac{1}{2}[/tex]([tex]r^{2}[/tex]) θ
A=[tex]\frac{1}{2}[/tex]([tex]6^{2}[/tex]) [tex]\frac{165\pi }{180}[/tex].....................................plug in values
A= =[tex]\frac{1}{2}[/tex]([tex]36[/tex]) [tex]\frac{11\pi }{12}[/tex]..................................simplify
A=51.84 [tex]in^{2}[/tex].....................................solve and round.
5)
S=r θ
S=10([tex]\frac{145\pi }{180}[/tex]).........................................plug in values
S=10( [tex]\frac{29\pi }{36}[/tex]).........................................simplify
S=25.31 yd.......................................solve and round
A=[tex]\frac{1}{2}[/tex]([tex]r^{2}[/tex]) θ
A=[tex]\frac{1}{2}[/tex]([tex]10^{2}[/tex]) [tex]\frac{145\pi }{180}[/tex].......................................plug in values
A= =[tex]\frac{1}{2}[/tex]([tex]100[/tex]) [tex]\frac{29\pi }{36}[/tex]....................................simplify
A=126.54 [tex]yd^{2}[/tex].....................................solve and round.
6)
S=r θ
S=3([tex]\frac{65\pi }{180}[/tex])..........................................plug in values
S= 3( [tex]\frac{13\pi }{36}[/tex]).......................................simplify
S=3.4ft............................................solve and round
A=[tex]\frac{1}{2}[/tex]([tex]r^{2}[/tex]) θ
A=[tex]\frac{1}{2}[/tex]([tex]3^{2}[/tex]) [tex]\frac{65\pi }{180}[/tex].......................................plug in values
A= =[tex]\frac{1}{2}[/tex]([tex]9[/tex])[tex]\frac{13\pi }{36}[/tex]......................................simplify
A=5.11 [tex]ft^{2}[/tex].........................................solve and round.
Solve to find the value of ‘a’ 10a-2=4a-1a
Answer: a = [tex]\frac{2}{7}[/tex].
Step-by-step explanation:
10a - 2 = 4a - 1a
10a - 2 = 3a
10a -2 + 2 = 3a + 2
10a = 3a + 2
10a - 3a = 3a -3a + 2
7a = 2
a = [tex]\frac{2}{7}[/tex]
The demand equation for a certain commodity is given by the
following equation.
p= =1/2+². - 20x + 1200, 0≤x≤ 120
Find x and the corresponding price p that maximize revenue.
The maximum value of R(x) occurs at x =0.
In the given demand equation, the value of x and corresponding price p that maximize revenue are x = 31.8 and p = $514.60.
How to Solve the Equation?To find the value of x and corresponding price p that maximize revenue, we first need to determine the revenue function. Revenue (R) is given by the product of price (p) and quantity demanded (x), so:
R(x) = xp
Substituting the given demand equation for p, we get:
R(x) = x(1/2 + ² - 20x + 1200)
Simplifying and rearranging:
R(x) = 1/2x + ²x - 20x² + 1200x
R(x) = -20x² + (1200 + ²)x + 1/2x
To find the value of x that maximizes revenue, we take the derivative of R(x) with respect to x, set it equal to zero, and solve for x:
dR/dx = -40x + (1200 + ²) + 1/2 = 0
Solving for x:
x = (1200 + ² + 1/2) / 40
x = 31.8
To find the corresponding price, we substitute this value of x back into the demand equation for p:
p = 1/2 + ² - 20(31.8) + 1200
p = $514.60
Therefore, the value of x and corresponding price p that maximize revenue are x = 31.8 and p = $514.60.
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Is -x=y proportional
Answer:
no
Step-by-step explanation:
Help please !!
The volume of a fixed amount of a gas varies directly as the temperature 7 and inversely as the pressure P. Suppose that I'-70 cm³ when 7-420 kelvin
kg
kg
and P-18-
Find the temperature when
is 60 cm and P-7
X
Answer:
420 K x (60 cm³ / 70 cm³) x (7 / 18) = 294.29 K
Step-by-step explanation:
Please help me find the answer
The name of this circle is Circle R.
The name of the radius is RB.
The name of the diameter is RI or RD.
What is a circle?In Mathematics, a circle can be defined as a closed, two-dimensional curved geometric shape with no edges or corners. Additionally, a circle refers to the set of all points in a plane that are located at a fixed distance (radius) from a fixed point (central axis).
What is a line segment?In Mathematics, a line segment can be defined as the part of a line in a geometric figure such as a triangle, circle, quadrilateral, etc., that is bounded by two (2) distinct points. Additionally, a line segment typically has a fixed length.
In this context, we have the following names;
Circle R.
Radius = RB.
Diameter = RD or RI.
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Match the term to the correct definition.
Question 2 options:
The military alliance of the USA, Britain, France, and other countries during WWII
the front lines where combat between opposing armies occurs
Germany, Italy, Japan
ideas or statements spread to promote one side's views and present an opposing side negatively.
an economic system characterized by private or corporate ownership of capital goods
1.
Propaganda
2.
Front
3.
free enterprise system
4.
Allied Forces
5.
Axis Powers
Answer:
Propaganda: ideas or statements spread to promote one side's views and present an opposing side negatively.
Front: the front lines where combat between opposing armies occurs.
Free enterprise system: an economic system characterized by private or corporate ownership of capital goods.
Allied Forces: the military alliance of the USA, Britain, France, and other countries during WWII.
Axis Powers: Germany, Italy, Japan.
prove or disprove the quadrilateral be low is a rectangle by using the concepts of slope and congruence
(100 points)
The proof that the quadrilateral is a rectangle is shown below
Proving or disproving that the quadrilateral is a rectangleFrom the question, we have the following parameters that can be used in our computation:
A = (-2, 3)
B = (-4, 1)
C = (2, -1)
D = (0, -3)
From the graph, we have the following lengths
AB = √8
CD = √8
BD = √32
AC = √32
The above shows that opposite sides are equal
From the graph, we have the following slopes
AB = 1
CD = 1
BD = -1
AC = -1
The above shows that opposite sides are have equal slopes and adjacent sides are their slopes to be opposite reciprocals
Hence, the quadrilateral is a rectangle
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two angles in a triangle are 38 and 122.write and sole an addition equation to determine the measure of the third angle c
The measure οf the third angle c is 20 degrees.
What is triangle angle sum prοperty?The triangle angle sum prοperty states that the sum οf the three interiοr angles οf a triangle is always equal tο 180 degrees. This prοperty hοlds true fοr all types οf triangles, whether they are acute, οbtuse, οr right-angled. The prοperty can be expressed as fοllοws:
Angle A + Angle B + Angle C = 180 degrees, where A, B, and C are the measures οf the interiοr angles οf the triangle.
38 + 122 + c = 180
Tο sοlve fοr c, we can simplify the left side οf the equatiοn:
160 + c = 180
Subtracting 160 frοm bοth sides, we get:
c = 20
Therefοre, the measure οf the third angle c is 20 degrees.
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Complete question:
Two angles in a triangle are 38° and 122°. Write the angle sum property to determine the measure of the third angle ∠c.
[Hint: Angle sum property = ∠a+ ∠b+ ∠c = 180°]
Use the drawing tools to form the correct answer on the provided graph.
Given the equation of the parabola x = -1/8(y - 3)^2 + 1, graph its focus and directrix.
Using drawing tools parabola graph has been made and uploaded in answer for the given equation.
Standard form of parabola having vertical axis of symmetry given by:
[tex](y - k)^2 = 4p(x - h)[/tex]
where (h, k): vertex and p: distance between the vertex and the focus or directrix.
Comparing the given equation[tex]x = -1/8(y - 3)^2 + 1[/tex] with the standard form, we can see that the vertex is at (1, 3) and p = -1/32.
Since the parabola opens to the left, the focus is to the left of the vertex at a distance of p = -1/32 units. Thus, the focus is located at (-1/32, 3).
The directrix is a vertical line to the right of the vertex and is located at a distance of p = -1/32 units. Thus, the directrix is the vertical line x = 33/32.
To graph the focus and directrix, we can plot the vertex (1, 3) on the coordinate plane, draw a horizontal line through the vertex, and then plot the focus (-1/32, 3) to the left of the vertex and the directrix x = 33/32 to the right of the vertex.
Note that the parabola [tex]x = -1/8(y - 3)^2 + 1[/tex] is symmetric with respect to the vertical line passing through the vertex.
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for the following right triangle. find the side length x. round your answer to the nearest hundreth.
Answer:
answer is 11.60
Step-by-step explanation:
formula is C^2 = a^2 + b^2
c = 13
a = 8
b = ?
169 = 64 + x^2
135 = x^2
I hope this helps!
Find the general solution to the differential eauation y 0 cos x = y sinx+sin 175x Assume x ∈ (−π/2,π/2), and use C (capital C) for your arbitrary constant.
The general sοlutiοn tο the differential equatiοn is [tex]\mathrm{y = Ce^{(sin(x))}}[/tex].
Describe Differentiatiοn?The derivative οf a functiοn represents the instantaneοus rate οf change οf the functiοn at a specific pοint. It is calculated by finding the limit οf the difference quοtient as the interval between twο pοints οn the functiοn apprοaches zerο. The derivative can be expressed as a functiοn οf the independent variable, and it prοvides valuable infοrmatiοn abοut the behaviοr οf the οriginal functiοn.
The prοcess οf differentiatiοn invοlves applying a set οf rules tο functiοns tο οbtain their derivatives. These rules include the pοwer rule, prοduct rule, quοtient rule, chain rule, and οther mοre advanced rules that are used tο differentiate mοre cοmplex functiοns.
Tο sοlve the given differential equatiοn, we can use the methοd οf integrating factοrs.
First, we can rewrite the equatiοn as:
y'cοsx = ysinx + sin(175x)
Next, we can multiply bοth sides by the integrating factοr, which is [tex]e^{(\int(cos(x) dx))} = e^{(\sin(x) + C)}[/tex], where C is a cοnstant οf integratiοn:
[tex]\mathrm {e^{(sin(x)) }y'cosx = e^{(sin(x))} ysinx + e^{(sin(x))}sin(175x) + Ce^{(sin(x))}}[/tex]
Nοw, we can recοgnize the left-hand side as the derivative οf [tex]e^{(sin(x))}y[/tex]:
[tex](e^{(sin(x))y)}' = e^{(sin(x))} y' + cos(x) e^{(sin(x))}y[/tex]
Substituting this intο the abοve equatiοn, we get:
[tex]\mathrm{(e^{(sin(x))y)}' = e^{(sin(x)) }ysinx + e^{(sin(x))}sin(175x) + Ce^{(sin(x))}}[/tex]
[tex]cos(x) e^{(sin(x))}y = e^{(sin(x))y)}'[/tex]
Separating variables and integrating bοth sides, we get:
[tex]\int e^{sin(x) }dy/y = \int cos(x) dx[/tex]
ln|y| + C = sin(x) + C'
where C' is anοther cοnstant οf integratiοn.
Therefοre, the general sοlutiοn tο the differential equatiοn is:
[tex]\mathrm{|y| = e^{(sin(x)) }e^{(C' - sin(x))}}[/tex]
[tex]\mathrm{y = \± e^{(C' - sin(x) + sin(x))}}[/tex]
[tex]\mathrm{y = Ce^{(sin(x))}}[/tex]
where C is an arbitrary cοnstant.
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Find the linear function with the following properties.
f(0)=8
Slope of f=−6
The linear function with the given properties is: f(x) = 8 - 6x.
What is linear function?A linear function is a mathematical equation that describes a straight line when graphed. It is the most basic type of function, where the output is equal to the input multiplied by a constant, known as the slope, and a constant added, known as the y-intercept. Linear functions are used to model relationships between two quantities and can be used to describe a wide range of phenomena, from physical properties to economic trends.
This linear function is used to calculate the value of a variable (x) when given the starting value of the function (f(0)) and the slope of the line (the rate at which the value of the function changes for each unit increase in x). In this case, the starting value of the function (f(0)) is 8, and the slope of the line is -6, meaning that the value of the function decreases by 6 for each unit increase in x. Therefore, the linear function that satisfies these properties is f(x) = 8 - 6x.
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