Answer:
Length between two points is given as √(x1-x2)²+(y1-y2)², where x and y are the coordinates of the points.
Distance = √(1-(-7))²+(-3-2)²
= √8²+(-5)²
= √64+25
= √89
= 9.43
= 9.4 (nearest tenth)
Find the equation of a parabola with a focus of (0, 9) and directrix y = –9.
Answer:
Step-by-step explanation:
Given that,
To find the standard form of the equation of the parabola with a focus at (0, 9) and a directrix y = -9.
What is a parabola?
A parabola is a cross-section cut out of the cone and represented by an equation
Focus of the prabola = (h , k + F ) = (0, 9)
Since the directrix, y = -9
F = -9
k + F = 9
k = 0
Vertex of the parabola = (h, k )
= (0, 0)
Standard equation of the parabola
( y - k ) = 4a (x - h)²
( y - 0 ) = 4a (x - 0)²
y = 4 * 9 x²
y = 36 x²
Thus, the required expression for the parabola with focus at (0, -9) and a directrix y = 9 is y = 36x².
Determine wether y varies directly with x. If so find the constant of variation k and write the equation
The given data is incorrect, k does not constitute a constant, but rather y does not varies directly to x.
What is defined as the direct variation?A simple connection between two variables is described by direct variation. If y=kx, we say it varies significantly with x (or even as x in some textbooks) for some constant k, known as the constant of variation or the constant of proportionality.Because y varies directly with x, it would fit a equation y=kx for each and every point with in set.We may have 11=7k for the initial point, implying that k=11/7.
Again for second point, we'd have 13=8k, which means k=13/8.
Using proportions, we can see that 11/7 doesn't really equal 13/8: cross multiply and you get 11*8=7*13, or 88=91.
Thus, this is incorrect, k does not constitute a constant, but rather y does not varies directly to x.
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The complete question is-
Determine whether y varies directly with x. If so, find the constant of variation k and write the equation.
x y
7 11
8 13
9 15
10 17
Find an equation for the perpendicular bisector of the line segment whose endpoints are (3, -8) and (7,2).
ANSWER
[tex]y=-\frac{2}{5}x-1[/tex]EXPLANATION
We want to find the equation of the perpendicular bisector of the line segment with the given endpoints.
To do this, we first have to find the slope of the given line, since the slope of a line perpendicular to a given line is the negative inverse of the slope of the line.
Then, we have to find the midpoint of the given line since the line is a bisector, it passes through the midpoint of the given line segment.
To find the slope of the line, apply the formula for the slope of a line:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]where (x1, y1) and (x2, y2) are the two endpoints of the line segment
Hence, the slope of the line is:
[tex]\begin{gathered} m=\frac{2-(-8)}{7-3}=\frac{2+8}{7-3} \\ m=\frac{10}{4} \\ m=\frac{5}{2} \end{gathered}[/tex]The negative inverse of this is:
[tex]\begin{gathered} -(\frac{1}{\frac{5}{2}}) \\ \Rightarrow-\frac{2}{5} \end{gathered}[/tex]To find the midpoint of the endpoints, apply the formula for midpoint:
[tex](\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]Hence, the midpoint of the given endpoints are:
[tex]\begin{gathered} (\frac{3+7}{2},\frac{-8+2}{2}) \\ \Rightarrow(\frac{10}{2},\frac{-6}{2}) \\ \Rightarrow(5,-3) \end{gathered}[/tex]Now, we have the slope and an endpoint of the perpendicular bisector.
To find the equation of the line, we have to apply the point-slope method:
[tex]y-y_1=m(x-x_1)[/tex]Therefore, the equation of the perpendicular bisector of the line segment is:
[tex]\begin{gathered} y-(-3)=-\frac{2}{5}(x-5) \\ y+3=-\frac{2}{5}x+2 \\ y=-\frac{2}{5}x+2-3 \\ y=-\frac{2}{5}x-1 \end{gathered}[/tex]What is the slope of a line perpendicylar to the line whose equation is 5x - 6y = 30 Fully simplify your answer.
Answer:
-6/5
Step-by-step explanation:
[tex]5x-6y=30 \\ \\ 6y-5x=-30 \\ \\ 6y=5x-30 \\ \\ y=\frac{5}{6}x-5[/tex]
Perpendicular lines have negative reciprocal slopes, so the answer is -6/5.
Samantha borrowed money to buy lawn equipment to start her new lawn service business. She borrowed $800 for 9 months and paid $70.50 in interest. What was the rate of interest.
Using simple interest, the rate of interest on the amount that Samantha borrowed to buy the law equipment is 11.75%.
What is the simple interest?A simple interest system does not accumulate (compound) interest on both the principal and interest, unlike compound interest.
The simple interest uses the following formula, Interest = Principal x Rate x Period.
This simple interest formula can be reversed to find either the principal, rate, or period, as the case may be.
The loan amount = $800
Period of loan = 9 months
Total interest paid = $70.50
Rate of interest = (Interest × 100)/(Principal × Time)
= 11.75% ($70.50 x 100)/($800 x 9/12)
Check:
Interest = $70.50 ($800 x 11.75% x 9/12)
Thus, Samantha's interest rate on the loan is 11.75%.
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To determine the value of tangent of 7 times pi over 8, which identity could be used?
Given:
[tex]\tan \frac{7\pi}{8}[/tex]To Determine: The identity that is equivalent to the given tangent
Note that, the identity rule below would be applied
[tex]\tan \frac{\alpha}{2}=\sqrt[]{\frac{1-\cos \alpha}{1+\cos \alpha}}[/tex]Also,
[tex]\tan \frac{\alpha}{2}=\frac{\sin \alpha}{1+\cos \alpha}[/tex]And also,
[tex]\tan \frac{\alpha}{2}=\frac{1-\cos \alpha}{\sin \alpha}[/tex]From the given tangent, we can re-write it as below:
[tex]\begin{gathered} \tan \frac{7\pi}{8}\cong\tan \frac{\frac{7\pi}{4}}{2} \\ \text{Note} \\ \frac{7\pi}{8}=\frac{\frac{7\pi}{4}}{2} \end{gathered}[/tex]Therefore:
[tex]\tan \frac{\frac{7\pi}{4}}{2}=\sqrt[]{\frac{1-\cos\frac{7\pi}{4}}{1+\cos\frac{7\pi}{4}}}[/tex]Also:
[tex]\tan \frac{\frac{7\pi}{4}}{2}=\frac{\sin \frac{7\pi}{4}}{1+\cos \frac{7\pi}{4}}[/tex]And also,
[tex]\tan \frac{\frac{7\pi}{4}}{2}=\frac{1-\cos \frac{7\pi}{4}}{\sin \frac{7\pi}{4}}[/tex]It can be observed from the option provided, the correct options is
I and III only
We have to find two partial products to add 513×46
Answer
Explanations:
The product of two integers using the partial product is expressed using the distributive law as shown:
[tex]513\times46=(500+13)\times(40+6)[/tex]Expanding the result using the distributive law as shown;
[tex]undefined[/tex]Look at the table. Is F(x) an exponential function? If so, Identify the base. if not, Why not?
ANSWER
YES, the base is 4 ......option B
NEED HELP WITH THIS MATH QUESTION QUICK!
Answer:
[tex]\textsf{The quotient is $\boxed{10}\;x+\boxed{16}$}[/tex]
[tex]\textsf{The remainder is $\boxed{28}\:x^2+\boxed{10}\:x+\boxed{22}$}[/tex]
Step-by-step explanation:
Definitions
Dividend: The polynomial which has to be divided.
Divisor: The expression by which the dividend is divided.
Quotient: The result of the division.
Remainder: The part left over.
Long Division Method of dividing polynomials
Divide the first term of the dividend by the first term of the divisor and put that in the answer.Multiply the divisor by that answer, put that below the dividend and subtract to create a new polynomial.Repeat until no more division is possible.Write the solution as the quotient plus the remainder divided by the divisor.Given:
[tex]\textsf{Dividend}: \quad 10x^4-14x^3-10x^2+6x-10[/tex]
[tex]\textsf{Divisor}: \quad x^3-3x^2+x-2[/tex]
Therefore:
[tex]\large \begin{array}{r}10x+16\phantom{)}\\x^3-3x^2+x-2{\overline{\smash{\big)}\,10x^4-14x^3-10x^2+6x-10\phantom{)}}}\\{-~\phantom{(}\underline{(10x^4-30x^3+10x^2-20x)\phantom{-b)}}\\16x^3-20x^2+26x-10\phantom{)}\\-~\phantom{()}\underline{(16x^3-48x^2+16x-32)\phantom{}}\\28x^2+10x+22\phantom{)}\\\end{array}[/tex]
Solution:
[tex]10x+16+\dfrac{28x^2+10x+22}{x^3-3x^2+x-2}[/tex]
[tex]\textsf{The quotient is $\boxed{10}\;x+\boxed{16}$}[/tex]
[tex]\textsf{The remainder is $\boxed{28}\:x^2+\boxed{10}\:x+\boxed{22}$}[/tex]
Choose the best answer. The diagonals of a rhombus:A. bisect each other and intersect at different anglesB. are the same length and intersect at a right angleC. are the same length and intersect at different anglesD. bisect each other at right angles
The property of a rhombus regarding the two diagonals states that they bisect each other at right angles.
Hence, option D is the correct answer.
A 25 ft ladder is leaning against a building.The base of the ladder is 6 ft away from the building.How high up is the ladder?
Applying the Pythagorean Theorem, the ladder's height from the ground is: 24.3 ft.
How to Apply the Pythagorean Theorem?If we know any two sides of a right triangle, the Pythagorean Theorem can be used to find the length of the third side, c, if c is the longest side, and a and b are the shorter sides of the right triangle, we ill have the equation:
c² = a² + b².
The ladder forms a right triangle with the wall of the building. Therefore:
c = length of the ladder = 25 fta = distance of base of the ladder from the building = 6 ftb = how high the ladder is up on the wall of the buildingSubstitute
25² = 6² + b²
b = √(25² - 6²)
b = 24.3 ft.
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I'll give brainliest!
The scientific notation is [tex]5.72 * 10^{6}[/tex]
Factor out 1 of the powers of 10
[tex]= (5.5 * 10^{6}/10^{6} + 2.2 * 10^{5}/10^{6}) * 10^{6}[/tex]
Perform division of exponents:
[tex](5.5 * 10^{0} + 2.2 * 10^{-1}) * 10^{6}[/tex]
Convert Scientific notations to real numbers:
[tex]= (5.5 + 0.22) * 10^{6}[/tex]
Combine real numbers:
[tex]= (5.72) * (10^{6})[/tex]
Convert to proper Scientific notations:
[tex]= 5.72 * 10^{6}[/tex]
Manually check answer
= (5.5 x 1000000) + (2.2 x 100000)
= 5500000 + 220000
= 5720000
[tex]= 5.72 * 10^{6}[/tex]
What are Scientific notations?
Scientific notation is a way to display extremely big or extremely small numbers in a more understandable way. We are aware that full numbers can go on forever, but we are unable to write such enormous figures on paper. Additionally, a simpler method of representation was required for the numbers that appear at the millions place following the decimal. This makes it challenging to express a small number of integers in their enlarged form. We thus employ scientific notations. Learn general forms for numbers as well.
Rules for Scientific Notation
We must adhere to the following rule in order to calculate the power or exponent of 10:
The base must always be 10.
Exponents that are non-zero integers must be either positive or negative in order to be used.
The coefficient's absolute value is more than or equal to 1, but it must be less than 10.
Positive and negative numbers, including whole and decimal values, can be coefficients.
The remaining significant digits of the number are represented by the mantissa.
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Under which transformation is size not preserved?A. reflectionB. dilationC. rotationD. translation
The philosophy of dilation is to resize uniformly the figure in question. Under this intuitive idea, dilation is the answer. Now, what does uniformly mean here?
It means that, as can be seen in the figure, the length of every side of the right triangle can be calculated by multiplying its corresponding side in the left ("small") triangle by a constant. However, this can be done from the right triangle to the left triangle; that's why I put a bidirectional arrow.
Finally, I want to give you some motivation for this concept: Every time you are resizing an image on your phone or in your computer, you're applying this concept.
could you please help me out with a question
of the circumference is 21.2, we get that the diameter is
[tex]d=\frac{21.2}{\pi}\approx6.75[/tex]and therefore the radius is
[tex]r=3.37[/tex]write the equation of the line that passes through the given points.
(4, 0) and (0, 2)
Answer:
Equation of line is given as y = mx + c, where m is the gradient and c is the y-intercept.
First find the gradient. Formula for gradient is given as (y2-y1)÷(x2-x1) or (y1-y2)÷(x1-x2).
Gradient = (2-0)÷(0-4) = -1/2
Equation of line is y = -1/2x + c
Substitute either one of the points into the equation to find c.
0 = -1/2(4) + c
c = 2
Hence, the equation of the line is y = -1/2x + 2.
Last year, Tammy opened an investment account with $8200 . At the end of the year, the amount in the account had decreased by 7.5% . How much is this decrease in dollars? How much money was in her account at the end of last year?
1. The decrease in dollars, based on a 7.5 percent decrease, is $615.
2. The balance in the account at the end of last year was $7,585.
What is a percentage?A percentage is a ratio or proportion of a variable in another.
For instance, the percentage decrease in the investment account was 7.5%, which in dollar terms amounts to $615.
The percentage reference gives an idea of the investment status and the fractional effect that the decrease had on it.
Initial investment = $8,200
Decrease in investment = 7.5%
Decrease in dollars = $615 ($8,200 x 7.5%)
Balance = $7,585 ($8,200 - $615)
Thus, Tammy's investment account decreased to $7,585 by $615, representing a 7.5% decrease.
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Find the values of x and y in the following right triangle. Enter square roots not decimals.
Recall the following trigonometric identities. If the legs of the right triangle have lengths a and b, the hypotenuse has length c, and the side a is adjacent to an angle θ, then:
[tex]\begin{gathered} \sin \theta=\frac{b}{c} \\ \cos \theta=\frac{a}{c} \end{gathered}[/tex]Then, for the given right triangle:
[tex]\begin{gathered} \sin (30º)=\frac{x}{8} \\ \cos (30º)=\frac{y}{8} \end{gathered}[/tex]Then, x and y are given by the expressions:
[tex]\begin{gathered} x=8\cdot\sin (30º)=8\cdot\frac{1}{2}=4 \\ y=8\cdot\cos (30º)=8\cdot\frac{\sqrt[]{3}}{2}=4\cdot\sqrt[]{3} \end{gathered}[/tex]Therefore, the answers are:
[tex]\begin{gathered} x=4\cdot\sqrt[]{3} \\ y=4 \end{gathered}[/tex]Write the equation of a line in the form y=Mx+b that passes through the point (3,6) and has a slope of -2/3. Sketch this line
What is the LCM of 9 and 15?
What is the LCM of 9 and 15?
we know that
9=(3^2)
15=(3)(5)
so
the LCM=(3^2)(5)=45
the answer is
LCM=453^2=3*3=9In a jar there are 65 red, 154 yellow and 70 green beads. You extract 54 beads at random.
Estimate the number of red beads obtained.
The number of red beads obtained from the jar is 12 beads.
How many red beads obtained?The number of red beads that can be obtained from the jar is the ratio of red beads to the total number of beads multiplied by the total number of beads picked at random.
Number of red beads obtained = (number of red beads / total number of beads) x number of beads picked
Total number of beads = 65 + 154 + 70 = 289
Number of red beads obtained = (65 / 289) x 54 = 12.14 ≈ 12 beads
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Fill in the blank to make equivalent rational expressions.
2/y^2=_/3y^5
The expression that can complete the blank is 6y^3
What are expressions?Expressions are mathematical statements that are represented by variables, coefficients and operators
How to complete the blanks?The expression is given as
2/y^2=_/3y^5
Replace the blank with x
So, we have
2/y^2 = x/3y^5
Multiply both sides of the equation by 3y^5
So, we have
x = 3y^5 * 2/y^2
Evaluate the products
x = 6y^3
This means that the blank is 6y^3
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Write an equation that says that the length of the green line is equal to the length of theblack line. Combine like terms
Explanation:
Length of green line = length of black line
Length of green line = 26
Total length of black line = h + h + *
Total length of black line = 2h + 8
Equate the two expressions
26 = 2h + 8
Collect the like terms
26 - 8 = 2h
18 = 2h
Isolate h by dividing both sides by 2
18/2 = 2h/2
h = 18/2
h = 9
The equation is 26 = 2h + 8
Find each indicated value or measure assume that all segments that appear to be tangent are tangent.
Answer:
6
Step-by-step explanation:
Using the intersecting chords theorem,
[tex]x^2=36 \\ \\ x=6 (x>0)[/tex]
1. It costs $9.81 for 9 pounds of grapefruit. How much would it cost for 1 pound of grapefruit?
Answer:
the anser is 1.09
Step-by-step explanation:
9.81 divided by 9 = 1.09
Carly has twice as many sisters as Connor.
Connor has twice as many sisters as Alicia.
Alicia has 3 sisters.
How many sisters does Carly have?
Answer:
12
Step-by-step explanation:
Alicia has 3 sisters.
Connor has twice as many sisters.
Twice means "two times".
Connor has 2×3, that is 6 sisters.
Carly has twice (two times) as many sisters as Connor.
Carly has 2×6, or 12 sisters.
Which one is the option to describe a piece wise function ?
.
A piece-wise function is a function that changes its value, based on the input
That is a function its range depends on the domain.
The correct answer is the third option
The first term of an Ap is 5 and the common difference is-3/2 Find the term whose value is 201/2
The term in the AP whose value is 201 / 2 is 71.3.
How to find the term in an arithmetic progression?The first term of an arithmetic progression is 5 and the common difference is - 3/2.
The formula of the arithmetic progression can be described as follows:
nth term = a + (n - 1)d
where
a = first termn = number of termsd = common differenceTherefore, using the arithmetic progression formula,
a = 5
d = 3 / 2
nth term = 201 / 2
let's find the term(n)
Therefore,
- 201 / 2 = 5 + (n - 1) - 3 / 2
- 201 / 2 = 5 - 3 / 2 n + 3 / 2
- 201 / 2 - 3 / 2 - 5 = - 3 / 2 n
- 107 = -3 / 2 n
-3n = - 214
n = 214 / 3
n = 71.3
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What are the coordinates of the foci of the conic section shown below?(y + 2)² /16 - (x − 3)² /9 = 1A. (3, -2±5)B.(-2+5,3)C. (-2,3±5)D.(-2+√7,3)
SOLUTION
Given the question on the question tab;
Explanation:
[tex]\frac{(y+2)^2}{16}-\frac{(x-3)^2}{9}=1[/tex][tex]h=3,k=-2,a=3,b=4[/tex][tex]The\text{ }standardform\text{ }is\frac{\text{ }\left(y+2\right)^2}{4^2}-\frac{\left(x - 3\right)^{2}}{3^{2}}=1.[/tex][tex]The\text{ }linear\text{ }eccentricity\text{ }is\text{ }c=\sqrt{b^{2} + a^{2}}=5.[/tex][tex]\begin{gathered} The\text{ first focus is:} \\ \left(h,k−c\right)=\left(3,−7\right). \\ The\text{ second focus is:} \\ \left(h,k+c\right)=\left(3,3\right) \end{gathered}[/tex]Final answer:
The graph of a line passes through the two points (-2, 1) and (2, 1). What is the equation of the line written in general form?answers:•x+y1=0•y-1=0•x-y+1 = 0
We will have the following:
First, we find the slope:
[tex]m=\frac{1-1}{2-(-2)}\Rightarrow m=0[/tex]So, the equation that represents the line is:
[tex]y=1[/tex]So:
[tex]y-1=0[/tex]***Explanation***
We are given the points (-2, 1) & (2. 1).
From this we cans see that the line passes both times through y = 1.
We also know that the slope (m) is given by:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]So the slope for this line will be:
[tex]m=\frac{1-1}{2-(-2)}\Rightarrow m=0[/tex]Now, we also know that the equation of the line in general form is given by:
[tex]y-y_1=m(x-x_1)[/tex]So, the equation of the line will be given by:
[tex]y-1=(0)(x+2)\Rightarrow y-1=0[/tex]So the function is a constant value, thus a horizontal line.
***Explanation for the general form of a line***
The general form of a line is given by:
[tex]y-y_1=m(x-x_1)[/tex]Here "x" & "y" are two variables; "x1" & "y1" are the component of a point that belongs to that lline and "m" is the slope of that line. We remember that (x1, y1) can be any point that belongs to the line.
So, we simply replace the values, in our case I willl solve for (-2, 1), but we could also use (2, 1), the solution at the end will be the same.
So:
[tex]y-(1)=(0)(x-(-2))\Rightarrow y-1=0\cdot x+0\cdot2[/tex][tex]\Rightarrow y-1=0[/tex]QUIK ANSWER PLEASE!!! Solve the equationy^3 - 27 = 9y^2 - 27y
The first step is to simplify both sides of the equation. The equation can be written as
y^3 - 3^3 = 9y(y - 3)
For the left hand side, we would apply the difference of two cubes formula. it is expressed as
x^3 - y^3 = (x - y)(x^2 + xy + y^2)
By comparing with the left hand side of the equation,
x = y and y = 3. It becomes
(y - 3)(y^2 + 3y + 3^2)
= (y - 3)(y^2 + 3y + 9)
The equation becomes
(y - 3)(y^2 + 3y + 9) = 9y(y - 3)
If we divide both sides of the equation by (y - 3), it becomes
(y - 3)(y^2 + 3y + 9)/(y - 3 = 9y(y - 3)/(y - 3)
y^2 + 3y + 9 = 9y
y^2 + 3y - 9y + 9 = 0
y^2 - 6y + 9 = 0
We would solve the quadratic equation by applying the method of factorisation. We would find two terms such that their sum or difference is - 6y and their product is 9y^2. The terms are - 3y and - 3y. The equation becomes
y^2 - 3y - 3y + 9
y(y - 3) - 3( y - 3) = 0
(y - 3)(y - 3) = 0
y - 3 = 0 twice
y = 3 twice