Answer:
D. x = 6
Step-by-step explanation:
Given equation:
[tex](x-2)^{\frac{1}{2}}+4=x[/tex]
Subtract 4 from both sides:
[tex]\implies (x-2)^{\frac{1}{2}}+4-4=x-4[/tex]
[tex]\implies (x-2)^{\frac{1}{2}}=x-4[/tex]
Square both sides:
[tex]\implies \left( (x-2)^{\frac{1}{2}}\right)^2=(x-4)^2[/tex]
[tex]\implies x-2=(x-4)^2[/tex]
Expand the brackets on the right side:
[tex]\implies x-2=(x-4)(x-4)[/tex]
[tex]\implies x-2=x^2-8x+16[/tex]
Subtract x from both sides:
[tex]\implies x-2-x=x^2-8x+16-x[/tex]
[tex]\implies -2=x^2-9x+16[/tex]
Add 2 to both sides:
[tex]\implies -2+2=x^2-9x+16+2[/tex]
[tex]\implies 0=x^2-9x+18[/tex]
[tex]\implies x^2-9x+18=0[/tex]
Factor the left side of the equation:
[tex]\implies x^2-6x-3x+18=0[/tex]
[tex]\implies x(x-6)-3(x-6)=0[/tex]
[tex]\implies (x-3)(x-6)=0[/tex]
Apply the zero-product property:
[tex]\implies x-3=0 \implies x=3[/tex]
[tex]\implies x-6=0\implies x=6[/tex]
Therefore, the solutions of the quadratic equation are:
[tex]x=3, \quad x=6[/tex]
Input both solutions into the original equation to check their validity:
[tex]\begin{aligned}x=3 \implies (3-2)^{\frac{1}{2}}+4&=3\\(1)^{\frac{1}{2}}+4&=3\\1+4&=3\\5&=3\end{aligned}[/tex]
[tex]\begin{aligned}x=6 \implies (6-2)^{\frac{1}{2}}+4&=6\\(4)^{\frac{1}{2}}+4&=6\\2+4&=6\\6&=6\end{aligned}[/tex]
Therefore, the only valid solution to the given equation is x = 6.
Answer:
The answer is D. 6
Step-by-step explanation:
Add both sides:
(x - 2)^1/2 + 4 + (-4) = x + (-4)
(x - 2)^1/2 = -4
Solve exponent:
(x - 2)^1/2 = x - 4
((x - 2)^1/2)^2 = (x - 4)^2
x − 2 = x^2 − 8x + 16
x − 2 − (x^2 − 8x + 16) = x^2 − 8x + 16 − (x^2 − 8x + 16)
−x^2 + 9x − 18 = 0
(−x + 3)(x − 6) = 0
−x + 3 = 0 or x − 6 = 0
x = 3 or x = 6
Check the answers: (Plug them in to see what will work.)
x = 3 (won't work)
x = 6 (does work)
Therefore,
x = 6
Can someone answer this question for me please, I really need the help:
Answer:
4x² +28x +49
Step-by-step explanation:
You want a polynomial equivalent to the area of the figure shown.
PolynomialThe polynomial can represent the sum of three areas:
blue area + green area + yellow area
= 4x² +4(7x) +7²
= 4x² +28x +49
find the lateral area and surface area the lateral area of the prism is __in squared. the surface area of the prism is __ in squared
Answer
The lateral area of the prism is 900 in squared
The surface area of the prism is 960 in squared
Explanation
Given:
The first side of the triangular base, a = 12 in
The second side of the triangular base, b = 13 in
The height of the prism, h = 30 in
What to find:
The lateral area and surface area of the prism.
Step-by-step solution:
The first step is to find the third side, c of the triangular base using Pythagoras rule.
[tex]\begin{gathered} b^2=c^2+a^2 \\ \\ 13^2=c^2+12^2 \\ \\ c^2=13^2-12^2 \\ \\ c^2=169-144 \\ \\ c^2=25 \\ \\ c=\sqrt{25} \\ \\ c=5\text{ }in \end{gathered}[/tex]Now, the next step is to calculate the lateral area of the prism using the formula below.
[tex]\begin{gathered} L.A=ha+hb+hc \\ \\ L.A=30\times12+30\times13+30\times5 \\ \\ L.A=360+390+150 \\ \\ L.A=900\text{ }in^2 \end{gathered}[/tex]The lateral area of the prism is 900 in squared
The final step is to calculate the surface area of the prism using the formula below.
[tex]S.A=Lateral\text{ }Area+Base\text{ }Area[/tex]The base area is
[tex]=2(\frac{1}{2}cb)=2(\frac{1}{2}\times5\times12)=2(\frac{60}{2})=60\text{ }in^2[/tex]Therefore, the Surface Area = (900 + 60) = 960 in squared
A rectangular safe can hold 5,184 cubic inches. Gold bars that are 3 inches by 6 inches by 2 inches fit to completely fill the safe. What is the volume of each gold bar?11 in.336 in.318 in.315 in.3
Given:
Rectangular safe hold 5184 cubic inches
Gold bar,
3 inches by 6 inches by 2 inches
Find-:
The volume of each gold bar
Explanation-:
The gold bar is in a rectangular shape
So volume is:
[tex]V=\text{ Length}\times\text{ Width}\times\text{ Height }[/tex]The dimension of the gold bar is 3 inches by 6 inches by 2 inches.
[tex]\begin{gathered} V=3\times6\times2 \\ \\ V=36\text{ in}^3 \end{gathered}[/tex]Volume of each gold bar is 36.
i just want the answers, i dont want an explanation.
Given:
Required: Evaluation
Explanation:
Use BODMAS rule to evaluate all
71.
[tex]\begin{gathered} 16+4(-3)-7=16-12-7 \\ =-3 \end{gathered}[/tex]72.
[tex]\begin{gathered} (-24)\div(4)+16(2)=-6+32 \\ =26 \end{gathered}[/tex]73.
[tex]\begin{gathered} -5-8+6(-3)=-13-18 \\ =-31 \end{gathered}[/tex]74.
[tex]\begin{gathered} 30-6\div3+4=30-2+4 \\ =32 \end{gathered}[/tex]75.
[tex]\begin{gathered} 21-45+8\div2=21-45+4 \\ =25-45=-20 \end{gathered}[/tex]Final Answer:
71 - E
72 - AE
73 - A
74 - BE
75 - B
Vector v has an initial point at (8, 4) and a terminal point at (−8, 10). What is -1/2 times vector v?
In order to find a vector multiplication by a scalar, we follow the rule:
[tex]u\ast(n_1,n_2)_\Rightarrow(un_{1,}un_2)[/tex]then, if we apply it to the vector v, we need to find the dimensions of the vector, for that we subtract the initial point from the terminal point,
[tex](-8-8,10-4)\Rightarrow(-16,6)[/tex]then, multiply by the scalar -1/2,
[tex]-\frac{1}{2}\ast(-16,6)\Rightarrow(8,-3)[/tex]Answer:
[tex](8,-3)[/tex]Does the graph represent a function? 4 5 6 7 -3+ no O yes
To prove:
The given graph is a function.
The given graph is a function if and only if no x value has more than one value of y, or we can say that a graph is a function iff no vertical line intersects the graph in more than one point.
Thus, by the given graph it is clear that for x = 7 their are two values for y that is y = -6 , -7
So, the given graph is not a function
Evaluate (You can use which ever first step you
selected above)
[tex] \frac{3 \times 3 \times 2^{ - 4} \times 2^{0} }{(3 \times 2 ^{ - 3}) ^{2} } \\ = \frac{3^{ 1+1 } \times 2^{ - 4 + 0} }{3 \times 2 ^{ - 3 \times 2} } \\ = \frac{ {3}^{2} \times {2}^{ - 4} }{3 \times {2}^{ - 6} } \\ = 3^{2 - 1} \times 2^{ - 4 - ( - 6)} \\ = 3 \times {2}^{ - 4 + 6} \\ = 3 \times {2}^{2} \\ = 3 \times 4 \\ = 12[/tex]
I USED THE LAWS OF EXPONENTS.
IF YOU DO NOT KNOW THEM PLEASE TELL ME AS SOON AS POSSIBLE TO ARRANGE FOR YOU.
f(x)=4x+6 and g(x)=1/2 * f(x)
What is the function rule for function g?
g(x)=
Answer: 2x + 3
Step-by-step explanation:
f(x) = 4x + 6
g(x) = 1/2 * f(x)
Plug f(x) into the g(x) equation.
g(x) = 1/2 * (4x + 6)
Now simplify,
g(x) = 2x + 3
2. A new car dealership invites 500 people to a promotional event. Only 330 people attend the event. What percentage of the people attend the event?
since
[tex]550X=330[/tex]then
[tex]X=\frac{330}{500}=0.66[/tex]then it is the 100x0.66= 66%
Use the formula h = 16t2, where t is time in seconds and h is the distance in feet traveled by a free-falling body or object to solve the following problem.
A diver dives from a cliff that has a height of 125 feet. Determine the time of the dive.
The dive lasted approximately
seconds.
(Type an integer or decimal rounded to the nearest tenth as needed.)
If a diver dives from a cliff that has a height of 125 feet, The dive lasted approximately 2.80 seconds
A diver dives from a cliff that has a height of 125 feet
The formula h = 16t², where t is time in seconds and h is the distance in feet traveled by a free-falling body or object
125 = 16t²
125/16 = t²
t = √(125/16)
t = √125/√16
t = 5√5 /4
t = 1.25 x 2.23
t = 2.795
t = 2.80
Therefore, if a diver dives from a cliff that has a height of 125 feet, The dive lasted approximately 2.80 seconds
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find the value of x and y that make the quadrilateral
The quadrilateral have equal opposite sides, for example:
Then, (4x + 6) must be equal to (7x - 3), and (4y - 3) is equal to (3y + 1), therefore
[tex]4x+6=7x-3[/tex]We can solve that equation for x
[tex]\begin{gathered} 4x+6=7x-3 \\ \\ 3x=9 \\ \\ x=\frac{9}{3}=3 \end{gathered}[/tex]Hence, x = 3. Now let's solve the other equation
[tex]\begin{gathered} 4y-3=3y+1 \\ \\ y=4 \end{gathered}[/tex]Then the value of y is 4.
Final answers:
x = 3
y = 4
Gravel is being dumped from a conveyor belt at a rate of 30 ft^3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?
The height of the pile increasing at the rate of 4.56 inches per minute when the pile is 10 ft high.
In given situation, the rate of change of height is equal to the rate of change of volume, divided by the base area.
First we find the base area.
base area = (π/4)d²
base area = (π/4)10²
base area = 25π ft²
base area ≈ 78.5 ft²
Then the rate of change of height would be,
(30 ft³/min)/(78.5 ft²)
≈ 0.38 ft/min
= 4.56 inches / minute
Therefore, the height of the pile increasing at the rate of 4.56 inches per minute when the pile is 10 ft high.
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Melissa brought 6 apples for $1.20 .if each apple cost the same amount, how much would 20 apples cost
The cost of 20 apples is such that the ratio of apples to cost maintained or each apple costs the same amount is $4.
What are the ratio and proportion?The ratio is the division of the two numbers.
For example, a/b, where a will be the numerator and b will be the denominator.
As per the given,
Melissa brought 6 apples for $1.20.
The ratio of cost to apple ⇒
1.20/6
Now let's suppose the cost of 20 apples is x dollars.
The ratio of cost to apple ⇒
x/20
Since the cost of each apple is the same, therefore, both ratios must be the same.
x/20 = 1.20/6
x = 20(1.20/6)
x = 4
Hence "The cost of 20 apples is such that the ratio of apples to cost maintained or each apple costs the same amount is $4".
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If P is the centroid of the triangle JKL, JK = 22, KN = 13, and OL = 18, find each measure.
Given
JK = 22
KN = 13
OL = 18
Procedure
Centroid formulation
KM = JK/2 = 22/2 = 11
KM = 11
NL = KN
NL = 13
KL = 2KN = 2(13) = 26
KL = 26
JO = OL = 18
JO = 18
JL = 2OL = 2(18)
JL = 36
A manufacturer of ski clothing makes ski pants and ski jackets. The profit on a pair of ski pants is $3 and on a jacket is $2. Both pants and jackets require the work of sewing operators and cutters. There are 60 minutes of sewing operator time and 48 minutes of cutter time available. It takes 8 minutes to sew one pair of ski pants and 4 minutes to sew one jacket. Cutters take 4 minutes on pants and 8 minutes on a jacket. The manufacturer wants to make a minimum of 4 ski pants and 2 ski jackets, Let x represent the number of ski pants. Let y represent the number of ski jackets.
For the sewing operator:
[tex]8x\text{ + 4y }\leq\text{ 60}[/tex]For cutter:
[tex]4x\text{ + 8y }\leq48[/tex]The two above equation is solve to obtain the graph
The solution of the graph is where the two lines meet
The coordinate is (10, 4), at the this point the maximum profit is made
So 10 ski pants and 4 ski jackets is made to get maximum profit
The profit made is given by
P = 3x + 2y
x =10 , y = 4
P = 3(10) + 2(4)
P = 30 + 8 = $38
it takes 126 cubes that have edge lengths of 1/3feet to completely fill this plastic bin what is the area of the base of bin
Answer: 1) Each of the 126 cubes has a volume = (1/3)3 = 1/3 × 1/3 × 1/3 = 1/27 cubic feet.
2) It takes 126 cubes to fill the bin, so the bin has a volume of 126 × 1/27 = 126/27 cubic feet. (Don't do this division yet - leave it as an improper fraction)
3) The volume of the bin is also given by the formula Volume = Area of Base × Height. Plus we know the volume is 126/27 cubic feet and the height is 2 1/3 = 7/3 of a foot, so:
Step-by-step explanation: Volume = Area of Base × Height
126/27 = Area of Base × 7/3
126/27 × 3/7 = Area of Base
The area of the base of bin is 2 feet².
What is Volume of Cube?The formula of volume of the cube is given by: Volume = a³, where a is the length of its sides or edges.
Given:
Edge = 1/3 feet
Volume of 1 cube= l³= (1/3)³ = 1/27
Totals cubes= 126
So, Volume of 126 cube= 126 x 1/27 = 126/ 27
Then, Volume of Plastic Bin= Area of base x height
126 / 27 = Area of base x 7/3
Area of base = 126 /27 x 3 /7
Area of base = 2 ft²
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14. Write a quadratic equation that cannot be factored.
Given:
Quadratic equation
To write a quadratic equation that cannot be factored or cannot be solved by factoring, we note first that factoring is the easiest method to solve a quadratic equation as long as the equation is easily factorable. If not, we'll need alternative approaches like employing the quadratic formula.
We also note that factoring cannot always be used to solve quadratic equations.
Below is an example of an equation that can be solve by factoring:
[tex]\begin{gathered} x^2-3x-10=0 \\ (x-5)(x+2)=0 \end{gathered}[/tex]Now, the equation that cannot be factored or cannot be solved by factoring is:
[tex]x^2+x-1=0[/tex]We cannot apply factoring to the above equation. Therefore, the answer is:
[tex]x^2+x-1=0[/tex]For j(x) = 3x − 1, find j of the quantity x plus h end quantity minus j of x all over h period a 3 to the power of the quantity x minus 1 end quantity times the quantity 3 to the power of h end quantity all over h b 3 to the power of the quantity x minus 1 end quantity times the quantity 3 to the power of h minus 1 end quantity all over h c 3 to the power of the quantity x minus 1 end quantity times the quantity 3 to the power of h plus 1 end quantity all over h d the quantity x minus 1 end quantity times the quantity 3 to the power of h plus 1 end quantity all over h
The resulting value of the function [j(x+h)-j(x)/h] is 3⁽ˣ⁻¹⁾ ([tex]3^{h}[/tex]-1)/h
What is an exponential function?An exponential function is defined as a function whose value is a constant raised to the power of an argument is called an exponential function.
It is a relation of the form y = aˣ in mathematics, where x is the independent variable
Given the function expressed as;
j(x) = 3⁽ˣ⁻¹⁾
Required value of the function [j(x+h)-j(x)]/h
We have to determine the function j(x+h) and j(x)
j(x+h) = [tex]3^{(x+h) -1}[/tex]
Substitute the values,
[j(x+h)-j(x)]/h = [ [tex]3^{(x+h) -1}[/tex] - 3⁽ˣ⁻¹⁾ ]/h
⇒ [ [tex]3^{(x-1) +h}[/tex] - 3⁽ˣ⁻¹⁾ ]/h
⇒ [ [tex]3^{(x-1)} \times3^{h}[/tex] - 3⁽ˣ⁻¹⁾ ]/h
⇒ 3⁽ˣ⁻¹⁾ ([tex]3^{h}[/tex]-1)/h
Hence, the equivalent value of the function is 3⁽ˣ⁻¹⁾ ([tex]3^{h}[/tex]-1)/h
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what is seven times negative three / 7(-3)
-21
1) Let's calculate this product:
7 (-3) Multiply 7 by -3
-21
2) So, seven times negative three is equal to -21 that is the same as adding seven times the number -3. Remember different signs in a product turns to out to a negative result.
You are about to enter an elevator with a weight capacity of 500 pounds. There are three people already in the elevator. The weights of the passengers are 107 pounds, 172 pounds and 129 pounds. Assuming that you weigh 136 pounds, is it safe for you to enter the elevator?
Given
capacity of 500 pounds.
weights of the passengers are 107 pounds, 172 pounds and 129 pounds.
you weigh 136 pounds
Find
is it safe for you to enter the elevator?
Explanation
as total pounds in the elevator = 107 + 172 + 129 = 408 pounds
total pounds included my weight = 408 + 136 = 544 pounds
capacity = 500 pounds
so , extra pounds = 544 - 500 = 44 pounds
so , no it is not safe for you to enter the elevator.
Final Answer
Therefore , No the weigh capacity would be exceeded by 44 pounds
Which of the following equations represents a line that is perpendicular toy=-3x+6 and passes through the point, (3, 2)?A. y=-3x+1B. y=(1/3x)+3C. y=(-1/3x)+1OD. y=(1/3x)+1
Step 1
For perpendicular lines
[tex]m_1=-\frac{1}{m_2}[/tex][tex]m_1=-3[/tex][tex]\begin{gathered} -3=-\frac{1}{m_2} \\ m_2=-\frac{1}{-3} \\ m_2=\frac{1}{3} \end{gathered}[/tex]Step 2
[tex]\begin{gathered} Given\text{ points \lparen3,2\rparen} \\ y=mx+b---\text{ Standard equation of a line} \\ y=\frac{1}{3}x+b \end{gathered}[/tex][tex]2=\frac{1}{3}(3)+b[/tex]Find b, the y-intercept
[tex]\begin{gathered} 2=1+b \\ b=2-1=1 \end{gathered}[/tex]Answer; Option D
[tex]y=\frac{1}{3}x+1[/tex]Find a_10 5, 12, 19, 26, 33...
Given,
The progression is 5, 12, 19, 26, 33...
The first term of the series is, a = 5.
The common difference of the series is,
d = 12 - 5 = 7
The 10th term of the series is,
[tex]\begin{gathered} a_{10}=a+(10-1)\times d \\ =5+9\times7 \\ =5+63 \\ =68 \end{gathered}[/tex]Hence, the 10th term of the series is 68.
A 220 m wire is cut into three pieces. The second piece is 2 times as long as the first. The third piece is 4 times as long as the second. How long is each piece?
The first piece = 20m
The second piece = 40m
The third pice = 8x = 160m
Explanation:Given:
Total length of wire = 220m
2nd piece = 2 times the 1st piece
3rd piece = 4 times the 2nd
To find:
The length of each of the piece
The wire was cut into 3:
1st piece + 2nd piece + 3rd piece = 220m
let the first length = x
2nd length = 2 (1st length) = 2(x)
2nd length = 2x
3rd length = 4(2nd piece) = 4(2x)
3rd length = 8x
[tex]\begin{gathered} x\text{ + 2x + 8x = 220} \\ simplify: \\ 11x\text{ = 220} \end{gathered}[/tex][tex]\begin{gathered} divide\text{ both sides by 11:} \\ \frac{11x}{11}=\frac{220}{11} \\ x\text{ = 20} \end{gathered}[/tex]The first piece = 20m
The second piece = 2x = 2(20) = 40m
The third pice = 8x = 8(20) = 160m
Find the equation of the circle (2, 3) and radius 4
Answer: [tex](\text{x}-2)^2+(\text{y}-3)^2 = 16[/tex]
=========================================================
Explanation:
The general circle template is
[tex](\text{x}-\text{h})^2+(\text{y}-\text{k})^2 = \text{r}^2[/tex]
(h,k) = center
r = radius
Assuming the center is (2,3) then we have h = 2 and k = 3. We also have a radius of r = 4.
[tex](\text{x}-\text{h})^2+(\text{y}-\text{k})^2 = \text{r}^2\\\\(\text{x}-2)^2+(\text{y}-3)^2 = 4^2\\\\(\text{x}-2)^2+(\text{y}-3)^2 = 16\\\\[/tex]
Which of the following statements best describe the branches of the hyperbola?Select one:a.It is not possible to determine how the branches of the hyperbola open.b.The branches of the hyperbola are positioned at a 30⁰ angle.c.The branches of the hyperbola open up and down.d.The branches of the hyperbola open side to side.
Answer:
Explanation:
The first step is to plot the graph of the parabola. It is shown below
Looking at the graph, the branches open up and down. Thus, the correct option is
c. The branches of the hyperbola open up and down.
Translate the following: "twice the sum of a number and 7
is negative fifteen"
A variable is a letter or term that represents an unknown number, value, or quantity. Variables are especially useful in algebraic expressions or algebra.
What is meant by variables?In the context of a mathematical problem or experiment, a variable is a quantity that can change. A variable is typically represented by a single letter. Variables are commonly represented by the letters x, y, and z. Any property, number, or quantity that can be measured or counted is defined as a variable.A variable is also referred to as a data item. Age, gender, business income and expenses, country of birth, capital expenditure, class grades, eye color, and vehicle type are all variables.In research, a variable is simply a person, place, thing, or phenomenon that you want to quantify in some way.let a number be x
then "twice the sum of a number be 2x
given
"twice the sum of a number and 7 is negative fifteen"
⇒ 2x + 7 = -15
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how many cubic blocks with a side length of 1/6 cm are needed to fill the volume of this prism?The answer choices are 4,8,9,32.
Given the figure, we can deduce the following information:
Height=1/2 cm
Length=1/2 cm
Width= 1/6 cm
To determine the number of cubic blocks with a side length of 1/6 cm needed to fill the volume of the given prism, we first note that a cube has equal sides. So, the dimensions of the cube must be:
Height= 1/6 cm
Length=1/6 cm
Width=1/6 cm
Next, we get the volume of the cube by using the formula:
[tex]\begin{gathered} Volume\text{ of the cube}=(Height)(Length)(Width) \\ =(\frac{1}{6})(\frac{1}{6})(\frac{1}{6}) \\ Simplify \\ =\frac{1}{216}\text{ }cm^3\text{ } \end{gathered}[/tex]Then, we get the volume of the given prism using the same formula:
[tex]\begin{gathered} Volume\text{ of the given prism}=(He\imaginaryI ght)(Length)(W\imaginaryI dth) \\ =(\frac{1}{2})(\frac{1}{2})(\frac{1}{6}) \\ =\frac{1}{24}\text{ }cm^3\text{ } \end{gathered}[/tex]Now, we find the number cubic blocks by using the formula:
[tex]\begin{gathered} Number\text{ }of\text{ cubic blocks}=\frac{Volume\text{ of the given prism}}{Volume\text{ of the cube}} \\ =\frac{\frac{1}{24}}{\frac{1}{216}} \\ Simplify \\ =9 \end{gathered}[/tex]Therefore, the answer is 9.
IXL Transversals of parallel lines: prove angle relationships 6QF for geometry, please help
4) [tex]m\angle XWY=m\angle HGY[/tex] (definition of congruent angles)
5) [tex]\angle HGY \cong \angle GTU[/tex] (corresponding angles theorem)
6) [tex]m\angle HGY=m\angle GTU[/tex] (definition of congruent angles)
7) [tex]\angle GTU[/tex] and [tex]\angle UTR[/tex] are supplementary (linear pair)
8) [tex]m\angle GTU+m\angle UTR=180^{\circ}[/tex] (definition of supplementary angles)
9) [tex]m\angle HGY+m\angle UTR=180^{\circ}[/tex] (substitution)
10) [tex]m\angle RTU+m\angle XWY=180^{\circ}[/tex] (substitution)
In a square box problem, the:Top number is the product of those left and rightBottom number is the sum of those at left and rightComplete this square box problem
Answer
Draw the box to complete the table.
Next,
To get the top, multiply both the left and the right = 11 x (-3)
= -33
To get the top, add both the left and the right = 11 + (-3) = 11 - 3 = 8
Find the standard form for the equation of a circle ( x − h )^2 + ( y − k )^2 = r^2 with a diameter that has endpoints ( − 4 , − 8 ) and ( 6 , − 4 ) .h=k=r=
Answer:
[tex](x-1)^2+(y+6)^2=29[/tex][tex]\begin{gathered} h=1 \\ k=-6 \\ r=\sqrt{29} \end{gathered}[/tex]Explanation:
Given:
Endpoints of the diameter of a circle as (-4, -8) and (6, -4)
To find:
Equation of a circle in standard form
The equation of a circle in standard form is generally given as;
[tex](x-h)^2+(y-k)^2=r^2[/tex]where (h, k) is the coordinate of the center of the circle and r is the radius of the circle.
We'll go ahead and determine the coordinates of the midpoint of the endpoints of the diameter which will be the coordinates of the center of the circle as seen below wi;
[tex]h=\frac{x_1+x_2}{2}=\frac{-4+6}{2}=\frac{2}{2}=1[/tex][tex]k=\frac{-8+(-4)}{2}=\frac{-12}{2}=-6[/tex]So the center of the circle has coordinates (1, -6)
Since h = 1, and k = -6 and we have that x1 = -4 and y1 = -8, we can go ahead and solve for r as seen below;
[tex]\begin{gathered} (-4-1)^2+(-8-(-6))^2=r^2 \\ (-5)^2+(-2)^2=r^2 \\ 25+4=r^2 \\ 29=r^2 \\ \therefore r=\sqrt{29} \end{gathered}[/tex]We can now write the equation of the circle in standard form as;
[tex]\begin{gathered} (x-1)^2+(y-(-6))^2=29 \\ (x-1)^2+(y+6))^2=29 \end{gathered}[/tex]