There are two methods of solving systems of equations:
substitutioneliminationSubstitution is where we substitute one equation into the other by isolating a certain variable, or a group of terms.
Elimination is where we subtract the two equations. Before doing this, we may have to multiply one equation by a certain number to make sure one variable cancels out.
Solving the QuestionWe're given the following equations:
[tex]x + 3y = 7[/tex][tex]2x + 4y = 11[/tex]Because they are organized in the same manner (i.e. x [operation] y [equals] number), it is easier for us to use elimination.
First, multiply the first equation by 2:
[tex]x + 3y = 7\\2(x + 3y) = 2(7)\\2x + 6y = 14[/tex]
Now, subtract the second equation from the one we just created:
[tex]\hspace{10}2x + 6y = 14\\-2x + 4y = 11\\\rule{67}{0.3}\\2y=3[/tex]
Solve for y:
[tex]y=\dfrac{3}{2}[/tex]
To solve for x, we can use substitution in the first equation:
[tex]x + 3y = 7\\\\x + 3(\dfrac{3}{2}) = 7\\\\x + \dfrac{9}{2} = 7\\\\x = 7- \dfrac{9}{2}\\\\x = 7- 4.5\\\\x = 2.5\\\\x=\dfrac{5}{2}[/tex]
Answer[tex]x=\dfrac{5}{2}[/tex]
[tex]y=\dfrac{3}{2}[/tex]
the graph of the derivative of a function f crosses the x-axis 3 times. what does this tell you about the graph of f
Answer:
the graph of f has 3 turning points
Step-by-step explanation:
The graph of a function has a turning point (local extreme) where the derivative is zero and changes sign.
DerivativeThe derivative of a function tells you the slope of that function's graph. When the derivative is positive, the function is increasing. When the derivative is negative, the function is decreasing.
Turning PointWhere the derivative changes sign from positive to negative, the graph of the function changes direction from increasing to decreasing. At the point where the derivative is zero (between positive and negative), the graph is neither increasing nor decreasing. A tangent to the function at that point is a horizontal line, and the function itself is at a local maximum, a turning point.
The reverse is also true. When the derivative changes sign from negative to positive, the function changes from decreasing to increasing. The turning point where that occurs is a local minimum.
3 CrossingsIf the derivative crosses the x-axis (changes sign) 3 times, then there are three local extremes in the graph of f. The graph of f has 3 turning points.
__
Additional comment
In the attached graph, we have constructed a derivative function (red) that crosses the x-axis 3 times. It is the derivative of f(x), which is shown in blue. The purpose is to show the local extremes of f(x) match the zero crossings of the derivative.
If x =2 y=3 z=4 solve the following
x² + y²
Answer:
13
Step-by-step explanation:
Given x = 2, y = 3 and z = 4. We'll evaluate the value of x² + y² with given condition.
First, remind that we are only given the expression of x-term and y-term only and therefore, z-term is not included - it's not to be considered.
Substitute x = 2 and y = 3 in the expression:
[tex]\displaystyle{2^2+3^2 = 4+9}\\\\\displaystyle{4+9 = 13}[/tex]
Hence, the value of x² + y² when x = 2 and y = 3 is 13.
Please let me know if you have any questions!
Select the correct answer. Which function has a domain of all real numbers?
y = -x + 5 y = -2(3x) ³ O A. B. OC. y = CD. y = (x + 2)² (2x) ³ (2x) - 7
The function that has a domain of all real numbers is:
A. [tex]y = 2x^{\frac{1}{3}} - 7[/tex].
What is the domain of a function?The domain of a function is the set that contains all possible input values for the function.
If a function has an even root, equivalent to an exponent of [tex]\frac{1}{n}[/tex] with n even, the domain is only positive values, while if the exponent is odd, the domain is all real values.
Researching the problem on the internet, the function with odd exponent is:
A. [tex]y = 2x^{\frac{1}{3}} - 7[/tex].
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Quadrilateral A'B'C'D' is the result of rotating quadrilateral ABCD by 60 about
the origin
y
0
Select all of the correct statements about the unchanged properties of quadrilateral
ABCD and quadrilateral A'B'C'D.
Choose all answers that apply.
O
B
3 4/5 6
BC and BC are both parallel to the y-axis
2B and B' have the same measures.
D and D' have the same coordinates.
None of the above
Activate
Answer: B
Step-by-step explanation:
A) False. Rotations do not preserve parallelism.
B) True. Rotations are rigid motions and thus preserve angle measure.
C) False. A rotation that changes the location of every point except for the center of rotation.
an equation for loudness, in decibles, is L=10log10 R where R is the relative intensity of the sound. Sounds that reach levels of 120 decibles or more are painful to humans what is the relative intensity of 120 decibles
Considering the logarithmic loudness equation, the relative intensity of 120 decibels is of [tex]R = 10^{12}[/tex].
What is the logarithmic loudness equation?The equation is:
[tex]L = 10\log{R}[/tex]
In which:
L is the loudness, in decibels.R is the relative intensity.For this problem, we have that L = 120, hence the relative intensity is found as follows:
[tex]120 = 10\log{R}[/tex]
[tex]\log{R} = 12[/tex]
[tex]R = 10^{12}[/tex]
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Show your steps in evaluating each of the following expressions. The steps count 4 points each, the answer is 1
The steps in evaluating each of the following expressions is shown below.
What is an expression?An expression is a mathematical equation which shows the relationship that exist between two or more numerical quantities or variables.
How to evaluate the given expressions?15 - 35/7 - 2 + 3 - 4
15 - (35/7) - 2 + 3 - 4 (bracket and division)
15 - 5 - 2 + 3 - 4 (regroup)
15 + 3 - 5 - 2 - 4 (subtract and add)
18 - 11 = 7.
Expression 2.10 + 2(9 - 5) - 16/18
10 + (2 × 4) - 8/9 (bracket and division)
10 + 8 - 8/9 (add)
18 - 8/9 (subtract)
162/9 - 8/9 = 17 1/9 or 154/9.
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Complete Question:
Show your steps in evaluating each of the following expressions. The steps count 4 points each, the answer is 1 point.
A. 15 - 35/7 -2 + 3 -4
B. 10 + 2(9 - 5) - 16/18
In a survey of 300 college graduates, 46% reported that they entered a profession closely related to their college major. If 9 of those survey subjects are randomly selected without replacement for a follow-up survey, what is the probability that 3 of them entered a profession closely related to their college major
Probability is 17.5% that 3 of them entered a profession closely related to their college
According to the statement
we have given that survey of 300 college graduates, 46% reported that they entered a profession closely and
We know that For each college graduate, there are only two possible outcomes. Either they have entered a profession closely related to their college major, or they have not. The probability of a college graduate having entered a profession closely related to their college major is independent of other college graduates, so we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
This is P(X = 3) when n = 8. and the value of p is 0.46 put the value in the binomial formula then the outcome of answer will 17.5%
So, Probability is 17.5% that 3 of them entered a profession closely related to their college
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Find the area of the region defined by the region defined by the inequality 2|x| + 3|y-1| ≤ 6
If [tex]x[/tex] and [tex]y-1[/tex] have the same sign, then either
[tex]x>0,y>1 \implies 2|x| + 3|y-1| = 2x + 3(y-1)=6 \implies 2x + 3y = 9[/tex]
or
[tex]x<0,y<1 \implies 2|x| + 3|y-1| = -2x - 3(y-1) = 6 \implies 2x + 3y = -3[/tex]
If [tex]x[/tex] and [tex]y-1[/tex] have opposite sign, then
[tex]x>0,y<1 \implies 2|x| + 3|y-1| = 2x - 3(y-1) = 6 \implies 2x -3y = 3[/tex]
or
[tex]x<0,y>1 \implies 2|x| + 3|y-1| = -2x + 3(y-1) = 6 \implies 2x-3y = -9[/tex]
This is to say that the region has boundaries given by these two sets of parallel lines, so we can equivalently describe the region with the set
[tex]R = \left\{(x,y) \mid -3\le2x+3y\le9 \text{ and } -9\le2x-3y\le3\right\}[/tex]
The area of [tex]R[/tex] is given by the double integral
[tex]\displaystyle \iint_R dx\,dy[/tex]
To compute the area, change the variables to
[tex]\begin{cases}u = 2x + 3y \\ v = 2x - 3y\end{cases} \implies \begin{cases}x = \frac14(u+v) \\ y = \frac16(u-v)\end{cases}[/tex]
The Jacobian for this transformation is
[tex]J = \begin{bmatrix} x_u & x_v \\ y_u & y_v \end{bmatrix} = \begin{bmatrix}1/4 & 1/4 \\ 1/6 & -1/6\end{bmatrix}[/tex]
with determinant [tex]\det(J) = -\frac1{12}[/tex]. Then the integral transforms to
[tex]\displaystyle \iint_R dx\,dy = \iint_R |J| \, du \, dv = \frac1{12} \int_{-3}^9 \int_{-9}^3 dv\, du[/tex]
which is 1/12 the area of a square with side length 12. Hence the integral evaluates to
[tex]\displaystyle \iint_R dx\,dy = \frac1{12}\times12^2 = \boxed{12}[/tex].
Solve the system of equations.
2x+y = 7
x - 2y = 6
Put your answer as a coordinate point, or use "no solution" or "infinitely many solutions"(aka "the set of all real numbers").
Answer:
Ans: (4,-1)
Step-by-step explanation:
Lets keep:
2x+y=7 --- equation 1
x - 2y=6 ----- equation 2
equation 2 x 2: 2x - 4y=12 -------equation 3
now subtract equation 1 from equation 3
2x - 4y = 12
(-) 2x + y = 7
----> -5y = 5 [ Divide both sides by -5 ]
------> y= -1
Substitute y= -1 into eqaution 1
----> 2x + -1 = 7 [ add 1 to both side]
----> 2x = 8 [Divide by 2 on both sids]
----> x=4
Ans: (4,-1)
Determine the (x, y) coordinates of the vertex of the parabola that represents each of the following functions:
The vertices of the parabolae are:
(h, k) = (- 3, - 1)(h, k) = (1, - 9)(h, k) = (4, 1)(h, k) = (3/4, 9/4)(h, k) = (- 3, 2)(h, k) = (0, 36)(h, k) = (7/2, 9/4)(h, k) = (5, - 1)(h, k) = (1, - 3)(h, k) = (- 1/2, 1)How to find the coordinates of the vertex of a parabola
Parabolae are represented by quadratic equations. In this problem we have parabolae in standard form and we need to determine its vertex form to find the needed information. Now we summarize the forms of quadratic equations:
Standard form
y = a · x² + b · x + c (1)
Vertex form
y - k = C · (x - h)² (2)
Please notice that (x, y) = (h, k) represents the vertex of the parabola.
To change quadratic equations from standard form into vertex form we need to apply algebraic handling:
y = x² + 6 · x + 8
y + 1 = x² + 6 · x + 9
y + 1 = (x + 3)²
(h, k) = (- 3, - 1)
y = x² - 2 · x - 8
y + 9 = x² - 2 · x + 1
y + 9 = (x - 1)²
(h, k) = (1, - 9)
y = - x² + 8 · x - 15
y = - 1 · (x² - 8 · x + 15)
y - 1 = - 1 · (x² - 8 · x + 16)
y - 1 = - 1 · (x - 4)²
(h, k) = (4, 1)
y = - 4 · x² + 6 · x
y = - 4 · [x² - (3/2) · x]
y + (- 4) · (9/16) = - 4 · [x² - (3/2) · x + 9/16]
y - 9/4 = - 4 · (x - 3/4)²
(h, k) = (3/4, 9/4)
y = x² + 6 · x + 11
y - 2 = x² + 6 · x + 9
y - 2 = (x + 3)²
(h, k) = (- 3, 2)
y = - x² + 36
y - 36 = - x²
(h, k) = (0, 36)
y = - x² + 7 · x - 10
y = - (x² - 7 · x + 10)
y + (- 1) · (9/4) = - (x² - 7 · x + 49/4)
y - 9/4 = - (x - 7/2)²
(h, k) = (7/2, 9/4)
y = x² - 10 · x + 24
y + 1 = x² - 10 · x + 25
y + 1 = (x - 5)²
(h, k) = (5, - 1)
y = 2 · x² - 4 · x - 1
y = 2 · (x² - 2 · x - 1/2)
y + 2 · (3/2) = 2 · (x² - 2 · x + 1)
y + 3 = 2 · (x - 1)²
(h, k) = (1, - 3)
y = - 4 · x² - 2 · x
y = - 4 · [x² + (1/2) · x]
y + (- 4) · (1/4) = - 4 · [x² + (1/2) · x + 1/4]
y - 1 = - 4 · (x + 1/2)²
(h, k) = (- 1/2, 1)
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Simplify b^10/b^2
A. b^5
B. b^-5
C. b^8
D. b^-8
Answer:
C. b^8.
Step-by-step explanation:
b^10/b^2 We subtract the exponents:-
= b^(10-2)
= b^8.
b^8
Step-by-step explanation:
b^10/b^2 ,. b^10-2 ;. b^8
help i’ll give brainliest!
find the exact value of sin (x-y) if sinx=4/9 and siny=1/4
Answer:
sin(x - y) = 0.21
Step-by-step explanation:
we have the sin values which we need to get cos values
sin (A-B) = sin A cos B - sin B cos A
sin² A + cos² A = 1
sin x = 4/9
cos² x = 1 - sin² x = 1 - 16/81 = 65/81
cos² x = 65/81
cos x = √65/9
sin y = 1/4
cos² y = 1 - sin² y = 1 - 1/16 = 15/16
cos² y = 15/16
cos y = √15/4
sin(x − y) = sin x cos y - sin y cos x
sin(x - y) = 4/9 √15/4 - 1/4 √65/9
sin(x - y) = (4√15-√65)/36
sin(x - y) = 0.21
socratic Narad T
Answer: C
Step-by-step explanation:
on edg
Please help with this
Answer:
the answer is 8
Step-by-step explanation:
thats what i got
A sculpture is formed from a square-based pyramid resting on a cuboid. The base of the cuboid and the base of the pyramid are both squares of side 3 cm. The height of the cuboid is 8 cm and the total height of the sculpture is 15 cm. The total mass of the sculpture is 738 g. The cuboid-part of the sculpture is made of iron with density 7.8 g/cm³. The pyramid is made from copper. Calculate the density, in g/cm³, of the copper.
Answer:
7.8g/cm
Step-by-step explanation:
Instructions: Find the missing segment in the image below.
Answer: 7
Step-by-step explanation:
By the triangle proportionality theorem,
[tex]\frac{?}{21}=\frac{3}{9}\\\\?=7[/tex]
F(x)= x(x+3)(x+1)(x-4) has zeros at x=-3
Answer: C) Sometimes positive; sometimes negative
============================================================
Explanation:
Pick a value between x = -1 and x = 0. Let's say we go for x = -0.5
Plug this into f(x)
f(x) = x(x+3)(x+1)(x-4)
f(-0.5) = -0.5(-0.5+3)(-0.5+1)(-0.5-4)
f(-0.5) = -0.5(2.5)(0.5)(-4.5)
f(-0.5) = 2.8125
We get a positive value.
This shows that f(x) is positive on the region of -1 < x < 0
----------------
Now pick a value between x = 0 and x = 4. I'll use x = 1
f(x) = x(x+3)(x+1)(x-4)
f(1) = 1(1+3)(1+1)(1-4)
f(1) = 1(4)(2)(-3)
f(1) = -24
Therefore, f(x) is negative on the interval 0 < x < 4
----------------
In short, f(x) is both positive and negative on the interval -1 < x < 4
It's positive when -1 < x < 0
And it's negative when 0 < x < 4
Answer:
sometimes positive sometimes negative
Step-by-step explanation:
I did it on Khan Academy
determine the fall (drop) on 20' of drainage piping with a 1/4" per ft. grade
The Fall (drop) of drainage piping is: A. 5".
Fall (drop) of drainage pipingIn order to determine the fall (drop) of drainage piping we would need to multiply the drainage piping by the inch per feet.
Hence:
Using this formula
Fall (drop) of drainage piping = Drainage piping× inch per feet
Where:
Drainage piping=20
Inch per feet=1/4"
Let plug in the formula
Fall (drop) of drainage piping =20×1/4"
Fall (drop) of drainage piping =5"
Therefore the correct option is A. 5".
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CAN SOMEONE SHOW ME HOW TO DO THIS PLEASE!
3.5 ft If the ADA guidelines state that a wheelchair ramps angle of elevation must equal 4.8°, would a ramp with the following dimensions be up to code? Show your work and explain. (Picture is
not drawn to scale.)
40 ft
Ꮎ°
Answer:
∠α = 5.001°
a easy calculator for this is:
https://www.calculator.net/right-triangle-calculator.html?av=3.5&alphav=&alphaunit=d&bv=40&betav=&betaunit=d&cv=&hv=&areav=&perimeterv=&x=57&y=16
Find the difference. express the answer in scientific notation. (5.29 times 10 superscript 11 baseline) minus (3.86 times 10 superscript 11 baseline)
The difference between (5.29 times 10 superscript 11 baseline) minus (3.86 times 10 superscript 11 baseline) is 1. 43 × 10^11
How to determine the notation
Given the expression
(5. 29 × 10^11) - (3. 86 × 10 ^11)
First, find the common factor
10^11 ( 5. 29 - 3. 86)
Then substract the values within the bracket
10^11 (1. 43)
Multiply with the factor, we have
⇒1. 43 × 10^11
Thus, the difference between (5.29 times 10 superscript 11 baseline) minus (3.86 times 10 superscript 11 baseline) is 1. 43 × 10^11
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Answer:
A
Step-by-step explanation:
Find the difference. Express the answer in scientific notation.
(5.29 times 10 Superscript 11 Baseline) minus (3.86 times 10 Superscript 11 Baseline)
1.43 times 10 Superscript 11
9.15 times 10 Superscript 11
1.43 times 10 Superscript 22
9.15 times 10 Superscript 22
The number 321.8 is 34% of x. What is the value of x rounded to the nearest whole number?
Answer:
946
Step-by-step explanation:
Let's make an equation :
34% of x = 321.8
Covert 34% into decimal by dividing by 100 :
34 ÷ 100 = 0.34
Rewrite equation with decimal form :
0.34x = 321.8
Divide both sides by x to make x the subject :
x = 321.8 ÷0.34
x = 946.470588235
To the nearest whole number will be 946 as 4 rounds it down
So our final answer will be 946
Hope this helped and have a good day
[tex] {\qquad\qquad\huge\underline{{\sf Answer}}} [/tex]
Here we go ~
[tex]\qquad \sf \dashrightarrow \: \dfrac{34}{100} \sdot x = 321.8[/tex]
[tex]\qquad \sf \dashrightarrow \: x = \cfrac{(321.8) \sdot(100)}{34} [/tex]
[tex]\qquad \sf \dashrightarrow \: x = 946.47[/tex]
Round off to nearest whole number :
[tex]\qquad \sf \dashrightarrow \: x = 946[/tex]
Solve kx-2=7 for x. A. x=5/k B. x=9k C. x=9-k D. x=9/k
Answer:
the answer is D- 9/k
Answer:
D- 9/k
Step-by-step explanation:
The ratio of two numbers is 2/3, and their sum is 535. One of the numbers is:
(Select one)
A. 242
B. 321
C. 667
D. 408
One of the numbers would be 321. Hence option B is true.
Used the concept of the Number system that states,
A writing system used to express numbers is known as a number system. It is the mathematical notation used to consistently express the numbers in a particular set using digits or other symbols.
Given that,
The ratio of the two numbers is 2/3, and their sum is 535.
Let us assume that,
The two numbers are x and y.
Hence we have;
[tex]\dfrac{x}{y} = \dfrac{2}{3}[/tex] .. (i)
And, [tex]x + y = 535[/tex] .. (ii)
From equation (i);
[tex]\dfrac{x}{y} = \dfrac{2}{3}[/tex]
[tex]x = \dfrac{2y}{3}[/tex]
Substitute the above value of x in (ii);
[tex]x + y = 535[/tex]
[tex]\dfrac{2y}{3} + y = 535[/tex]
[tex]2y + 3y = 535 \times 3[/tex]
[tex]5y = 1605[/tex]
[tex]y = 321[/tex]
From equation (i);
[tex]x = \dfrac{2y}{3}[/tex]
[tex]x = \dfrac{2\times 321}{3}[/tex]
[tex]x = 214[/tex]
Therefore, the number is 321. So option B is true.
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Look at the pic and show your work
We kindly invite to check the image attached below to see the representation of the exponential function. This function shows exponential growth.
How to graph exponential functions
In this occasion we must plot the graph of exponential functions of the form:
y = a · bˣ (1)
Where:
a - Initial valueb - Base of the functionx - Independent valuey - Dependent valueFirst, we need to follow this procedure to create the graph of the curve on Cartesian plane:
Evaluate the function at every x-value.Fill the blanks on table.Mark the rectangular points (x, y) on the Cartesian plane.Match the points.Therefore, we build the exponential curve with the help of a graphing tool (i.e. Desmos), whose result is shown in the image attached below.
From (1) we must understand that exponential functions report growth for b > 1 and decay for 0 < b < 1. Thus, the exponential function y = 3ˣ shows exponential growth according to graphical and analytical findings.
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Which quadratic function is in standard
form?
Answer:
h(x) = 2x² -8x -10
Step-by-step explanation:
In the US, a quadratic is in standard form when the terms are listed in order of decreasing degree.
In the UK, a quadratic is in standard form when it is written in vertex form.
__
USThe only function with terms in order of decreasing degree is ...
h(x) = 2x² -8x -10
UKAll of the functions except the last are written in vertex form:
f(x) = (x +1)² +0 . . . . . . . . would usually be written f(x) = (x +1)²y(x) = -(x -6)² +16g(x) = -3(x -3)² +4__
Additional comment
Since the question asks about one function, we assume it is from the perspective of the US understanding of standard form. The point here is that "standard form" may vary from one tradition to another. (The "standard form" for numerical values varies by tradition, as well.)
ا سکول کے پرنسپل کے نام سکول میں ہم نصابی سرگرمیوں کا اہتمام کرنے کی درخواست ھیے ۔
Answer:
Step-by-step explanation:
a pot contains 5 black beads and 7 white beads.another pot contain 6 white beads. if one beads is drawn from each pot without looking, what will be the probability of getting atleast one white beads?
Step-by-step explanation:
is there something missing in the question ?
or is this meant to be a trick question ?
let me repeat :
pot1 contains 5 black and 7 white beads
pot2 contains 6 white beads (and nothing else, right ?)
then one bead is drawn from each pot.
so, 1 bead from pot1, and 1 bead from pot2.
the bead from pot1 is either white or black.
the bead from pot2 is white for sure.
so, the probability to get at least 1 white bead is 100% or 1, as there will be always the white bead taken from pot2.
The area of a sector of a circle of radius 8cm is 45cm². Find the size of the angle subtended at the centre of the circle, correct to one decimal place. (Take π = 22/7)
The size of the angle subtended at the center of the circle is θ = 80.5397°
What is the Area of the Sector?In circles, a sector is said to be a part of a circle made of the arc of the circle together with its two radii. This means that it is a portion of the circle formed by a portion of the circumference (arc) and radii of the circle at both endpoints of the arc.
The formula for Area of a sector is given as;
A = θ/360 x πr²
where;
θ is the central angle of the sector
r is radius
Given data ,
The area of a sector of a circle of radius 8cm is 45cm²
On simplifying , we get
A = θ/360 x πr²
45 = θ/360 ( 22/7 ) ( 8 )²
45 = ( θ/360 ) ( 201.1428 )
Divide by 201.1428 on both sides , we get
( θ/360 ) = 0.22372159
Multiply by 360 on both sides , we get
θ = 80.5397
Hence , the angle is θ = 80.5397°
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Determine the number of terms in the sequence: –45, –32, –19, –6, ..., 124.
Step-by-step explanation:
[tex] - 32 + 45 = 13 \\ t_{n} = ( a_{1} + (n - 1)d) \\ \\ d = 13 \: \: a_{1} = - 45[/tex]
[tex] t_{n} = - 45 + (n - 1)13 = = = > \\ - 45 + 13n - 13 = = = > \\ t_{n} = 13n - 58[/tex]
and now
[tex]124 = 13n - 58 = = = > \\ 182 = 13n = = = > n = 14[/tex]
The number of terms in the sequence: –45, –32, –19, –6, ..., 124 = 9.
The common difference is -45 - (-32)= 13
d = 13.
What is arithmetic progress?AP is a sequence of numbers in order, in which the difference among any two consecutive numbers is a constant cost. it's also referred to as mathematics series.
using arithmetic progress:-
last term = (n-1)d
first term(a) = –45
term = a + (n-1)d
there is a difference of 13, so the sequence will be
–45, –32, –19, –6,7, 20, 33, 46, 59, 72, 85, 98, 111, 124.
∴ number of terms = 9
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In Circle P, chord AB measures 4x - 6 centimeters and chord CD measures 6x - 12
centimeters. If Segment AB and Segment CD are each 4 centimeters from P, find AP.
Answer:
5 cm
Step-by-step explanation:
When two chords are equidistant from the center of the circle, the two chords have equal length. Therefore, the length of chord AB, 4x - 6, is equal to the length of chord CD, 6x - 12.
4x - 6 = 6x - 12
6 = 2x
x = 3
Now that we know x = 3, we can substitute the value back into the original expression, 4x - 6, to find the length of chord AB.
AB = 4x - 6 = 4*3 - 6 = 12 - 6 = 6 cm
When measuring the distance between a line and a point, we create a segment through the point perpendicular to the line. In Geometry, we also learn that the perpendicular bisector of a chord in a circle contains the center of the circle.
In this case, the gray line perpendicularly bisects segment AB. PX is 4 cm long and AX is 3 cm long (because AX is half the length of AB). Notice that triangle AXP is a right triangle, so we can use Pythagorean Theorem to find AP.
[tex]AP^{2} = AX^{2} +PX^{2}[/tex]
[tex]AP = \sqrt{3^{2} +4^{2} }[/tex]
[tex]AP =\sqrt{9+16}[/tex]
[tex]AP = \sqrt{25}[/tex]
[tex]AP = 5[/tex]