Step-by-step explanation:
so, "- 1" is also part of the square root ?
sqrt(x - 1) = x - 3
it is clear that for any value x < 1 we have no solution in R (as this makes the argument of the square root negative, and there is so real number solution for the square root of negative numbers).
now square the whole equation.
x - 1 = (x - 3)² = x² - 6x + 9
x² - 7x + 10 = 0
the general solution for quadratic equations is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
a = 1
b = -7
c = 10
x = (7 ± sqrt(49 - 4×1×10))/(2×1) =
= (7 ± sqrt(49 - 40))/2 = (7 ± sqrt(9))/2
x1 = (7 + 3)/2 = 10/2 = 5
x2 = (7 - 3)/2 = 4/2 = 2
x2 is probably (given the answer options) not a valid solution for the original problem, as it represents the negative solution of sqrt(x - 1).
sqrt(2 - 1) = 2 - 3
± 1 = -1
remember, every square root has always 2 solutions : a positive and a negative one.
your teacher clearly only wanted the positive solution, which is x1 = 5.
so, yes,
C. x = 5
is the correct answer.
but please send your teacher my regards and comments. he/she has to state that only the positive solution to the square root is required/allowed.
because, formally, also x = 2 is a valid solution.
and therefore, C. AND D. are correct answers !!!!
A cone will be inscribed in a sphere of radius 11 in.
Part A: Find the exact dimensions of the cone of largest volume that will fit inside the sphere. (20 points)
Part B: Find the maximum volume cone to the nearest tenth. (20 points)
The exact dimensions of the cone of largest volume that will fit inside the sphere will be 14.67 inches and 22✓2/3 inches.
How to calculate the dimensions?From the information given, the height of the cone will be:
h = 11 + x
h = 11 + 3 2/3
h = 14 2/3 inches
The radius of the base of the cone will be:
= ✓121 - ✓121/9
= 2✓2(11)/3
= 22✓2/3
Therefore, the dimensions of the cone aren't 14.67 inches and 22✓2/3 inches.
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What is the value of e^in7x?
1
7e
7x
7
Answer:
[tex]7x[/tex]
Step-by-step explanation:
[tex]\text{Let,}~~\\\\~~~~~~~y = e^{\ln 7x}\\\\\implies \ln y = \ln \left(e^{\ln 7x} \right)\\\\\implies \ln y = \ln (7x) \ln e\\\\\implies \ln y = \ln (7x)\\\\\implies y = 7x\\\\\implies e^{\ln 7x} = 7x[/tex]
The area of a rectangular shop in the mall is 160 square meters. The perimeter is 52 meters. What are the dimensions of the shop?
Answer:
width = 10 m
length = 16 m
Step-by-step explanation:
Formula
[tex]\textsf{Area of a rectangle} = wl[/tex]
[tex]\textsf{Perimeter of a rectangle}=2(w+l)[/tex]
(where [tex]w[/tex] is width and [tex]l[/tex] is length)
Given:
Area = 160 m²Perimeter = 52 mSubstituting the given values into the formulae to create two equations:
[tex]\textsf{Equation 1}: \quad wl=160[/tex]
[tex]\textsf{Equation 2}: \quad 2(w+l)=52[/tex]
Rearranging Equation 1 to make w the subject:
[tex]\implies w=\dfrac{160}{l}[/tex]
Substituting expression for w into Equation 2 and solving for [tex]l[/tex]:
[tex]\implies 2\left(\dfrac{160}{l}+l\right)=52[/tex]
[tex]\implies \dfrac{160}{l}+l=26[/tex]
[tex]\implies 160+l^2=26l[/tex]
[tex]\implies l^2-26l+160=0[/tex]
[tex]\implies l^2-10l-16l+160=0[/tex]
[tex]\implies l(l-10)-16(l-10)=0[/tex]
[tex]\implies (l-10)(l-16)=0[/tex]
[tex]\implies l=10, 16[/tex]
According to Equation 1:
If length = 10 m ⇒ width = 16 mIf length = 16 m ⇒ width = 10 mAs width < length, the dimensions of the shop are:
width = 10 mlength = 16 mThe diagram shows 3 identical circles inside a rectangle.
Each circle touches the other two circles and the sides of a rectan
as shown in the diagram.
The radius of each circle is 2 mm.
Work out the exact area of the rectangle.
Give your answer in the form a√3 + b
where a and b are integers.
Based on the calculations, the exact area of this rectangle is equal to 16√3 + 32 mm².
How to calculate the area of a rectangle?Mathematically, the area of a rectangle can be calculated by using this formula:
Area = Length × Width
Note: Radius (R) is equal to 2 mm.
Joining the center of the circles would form an equilateral triangle with side length of 2R. Also, the height of the equilateral triangle is given by:
Height = √2R² - R²
Height = √3R
For the length and width of this rectangle, we have:
Length = R + 2R + R
Length = 4R
Width = 2R + height of equilateral triangle
Width = 2R + √3R
Width = (2 + √3)R
Therefore, the area of a rectangle becomes:
Area = 4R × (2 + √3)R
Area = 4(2 + √3) × R²
Area = 4(2 + √3) × 2²
Area = (8 + 4√3) × 4
Area = 16√3 + 32 mm² ≡ a√3 + b
In conclusion, a = 16 and b = 32.
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X - 4.5y=20.5 - X + 4.7y= -21.5
The value of x and y is -0.0434 and -4.5652.
What is equation?equation, statement of equality between two expressions consisting of variables and/or numbers.
given:
x - 4.5y=20.5
x + 4.7y= -21.5
Solving above two equation we have:
20.5+4.5y=-21.5-4.7y
42= -9.2y
y= -42/9.2
y= -4.5652
x=-0.0434
Hence, the value of x and y is -0.0434 and -4.5652.
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A 6 foot shadow is cast by a 10 foot lamp post. What is the angle of elevation of the sun?
Let that be theta
tanØ=Perpendicular/BaseHere perpendicular is lamp post and base is the shadow
tanØ=10/6tanØ=5/3tanØ=1.67Ø=59°Answer:
59.0° (nearest tenth)
Step-by-step explanation:
This can be modeled as a right triangle (see attachment).
To find the angle of elevation (marked as [tex]x[/tex] on the attached diagram), use the tan trig ratio:
Tan trigonometric ratio
[tex]\sf\tan(\theta)=\dfrac{O}{A}[/tex]
where:
[tex]\theta[/tex] is the angleO is the side opposite the angleA is the side adjacent the angleGiven:
[tex]\theta = x[/tex]O = 10A = 6Substitute the given values into the formula and solve for x:
[tex]\implies \sf\tan(\theta)=\dfrac{10}{6}[/tex]
[tex]\implies \sf\theta=\tan^{-1}\left(\dfrac{10}{6}\right)[/tex]
[tex]\implies \theta = \sf 59.03624...^{\circ}[/tex]
[tex]\implies \theta = \sf 59.0^{\circ}\:(nearest\:tenth)[/tex]
Solve. 5/8 +3/4 over -2/3 - 5/6
The solution of the fraction 5/8 +3/4 over -2/3 - 5/6 is -11/12
How to solve fractions5/8 + 3/4 ÷ -2/3 - 5/6
(5+6) / 8 ÷ (-4-5) / 6
= 11/8 ÷ -9/6
= 11/8 ÷ -3/2
= 11/8 × 2/-3
= 22/-24
= -11/12
Therefore, the solution of the fraction 5/8 +3/4 over -2/3 - 5/6 is -11/12
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Use this tax table to find how much tax you need to pay on a taxable income of $40,000.
If taxable
But not
The tax is:
income is over--
over-
$0
$7,825
10 percent of the
amount over $0
$7,825
$31,850
$782.50 plus 15 percent
of the amount over
7,825
$31,850
$77,100
$4,386.25 plus 25
percent of the amount
over 31,850
$77,100
$160,850 $15,698.75 plus 28
percent of the amount
over 77,100
$160,850
$349,700 $39,148.75 plus 33
percent of the amount
over 160,850
$349,700
no limit
$101,469.25 plus 35
percent of the amount
over 349,700
OA $6,423.75
OB. $8,150.50
OC. $12,536.25
Answer:
b
Step-by-step explanation:
Amount of tax you need to pay on a taxable income of $40,000 is,
= $6423.75
What is compound interest?It is defined as the interest on the principal value or deposit and the interest which is gained on the principal value in the previous year.
According to table, $40,000 falls in the 3rd category between $31,850 and $77,100
And, The tax is "$4,386.25 + 25% of amount over 31,850"
Hence, we have:
= $4,386.25 + 25% of (40,000 - 31,850).
= $4,386.25 + 0.25 x 8150
= $6423.75
Thus, Amount of tax you need to pay on a taxable income of $40,000 is,
= $6423.75
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An angle measure 20 more than the measure of its supplementary angle.What is the measure of each angle?
Based on the definition of supplementary angles, the measure of both angles are: 80° and 100°.
What are Supplementary Angles?The sum of angles that are supplementary to each other equals 180 degrees.
Let the measure of the supplementary angle be x.
The other angle will be: x + 20.
(x + 20) + x = 180
x + 20 + x = 180
2x + 20 = 180
2x = 180 - 20
2x = 160
x = 160/2
x = 80°
The other angle will be x + 20 = 80 + 20 = 100°.
The measure of the angles are: 80° and 100°.
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Solve for x: -24 = 6x
1/4
-4
4
- 1/4
Answer:
-4
Step-by-step explanation:
[tex]~~~~~-24 = 6x\\\\\\\implies -\dfrac{24} 6 = \dfrac{6x}6\\\\\\\implies x = -4[/tex]
Solve the radical equation.
√x − 1 + 5 = 9
Answer:
17
Step-by-step explanation:
square root of x-1 +5=9
square root of x-1 = 4multiple by power of 2 because square root is cancled by 2.
x-1=16,x=17
please help asap
3/2
3
7/2
Answer:
[tex] \frac{7}{2} [/tex]
Step-by-step explanation:
[tex] \frac{3! + 0!}{2! \times1! } = \frac{(3 \times 2 \times 1) + 1}{(2 \times 1) \times 1} = \frac{6 + 1}{2} = \frac{7}{2} [/tex]
A line that passes through the point (x,y) with a y-intercept of b and a slope of m can be represented by the equation y = mx + b.
Natasha drew a line on the coordinate plane that passes through the point (15,13) and has a y-intercept of -8. What is the slope, m, of the line?
A slope is also known as the gradient of a line. The slope of the line is 1.4.
What is Slope?A slope also known as the gradient of a line is a number that helps to know both the direction and the steepness of the line.
Natasha drew a line on the coordinate plane that passes through the point (15,13) and has a y-intercept of -8. Since the coordinate of a point on the line is known the value of the y-intercept is known, therefore, we can write,
13 = m15 - 8
13 + 8 = 15m
21 = 15m
m = 7/5 = 1.4
Hence, the slope of the line is 1.4.
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20. THOUGHT PROVOKING There are 100 coins in a bag.
Only one of them has a date of 2010. You choose
a coin at random, check the date, and then put the
coin back in the bag. You repeat this 100 times. Are
you certain of choosing the 2010 coin at least once?
Explain your reasoning.
Answer:
yes
Step-by-step explanation:
yes, because there is a 99/1 chance. also, you can put 50 2010 coins in.
If the cost of 3 notebooks is 30.30, then find the cost of 15 such notebooks.
Answer:
If the cost of 3 notebooks is 30.30, we can infer the cost of one notebook is 10.10.
Step-by-step explanation:
[tex]10.10 * 15 = x[/tex]
[tex]10.10 * 15 = 151.5[/tex]
Hi student, let me help you out! :)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
We are asked to find the cost of 15 notebooks, given that the cost of 3 notebooks is 30.30.
[tex]\triangle~\fbox{\bf{KEY:}}[/tex]
First of all, we need to find the cost of 1 notebook.So we divide the cost of three notebooks (30.30) by 3:
[tex]\star~\large\pmb{30.30\div3}[/tex]
[tex]\star~\large\pmb{10.10}[/tex]
That's the cost of 1 notebook. Now, multiply that times 15 (we're asked to find the cost of 15 notebooks).
[tex]\star~\large\pmb{10.10\times15}[/tex]
This gives us:
[tex]\star~\large\pmb{151.5}[/tex]
That's the cost of 15 notebooks if the cost of one notebook is 10.10.
Hope it helps you out! :D
Ask in comments if any queries arise.
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~Just a smiley person helping students out :)
A line connecting the points (-3,0) and (x, -6)has slope -1. Determine x.
Answer:
x=3
Step-by-step explanation:
We can use the slope formula to help complete this question
m = (y2-y1)/(x2-x1)
-1 = ( -6-0)/(x- -3)
-1 = -6/ (x+3)
Multiply each side by (x+3)
-x-3 = -6
Add 3 to each side
-x -3+3 = -6+3
-x =-3
Multiply each side by -1
x = 3
Answer:
x = 3
Step-by-step explanation:
The given slope and point can be used to write the equation of the line using the point-slope form. To find x for a given y, we can solve that equation.
__
setupThe point-slope form of the equation for a line is ...
y -k = m(x -h) . . . . . . line with slope m through point (h, k)
For (h, k) = (-3, 0), and m = -1, the equation is ...
y -0 = -1(x -(-3))
y = -x -3
We want x when y = -6, so this becomes ...
-6 = -x -3
solutionAdding x+6 to both sides of the equation, we get ...
(x +6) -6 = -x -3 +(x +6)
x = 3 . . . . . . . simplify
41 is a prime number found in the middle of a list of 7 consecutive numbers. find the 7 consecutive numbers
Answer: 29, 31, 37, 41, 43, 47, 53.
Step-by-step explanation:
Andrew is showing his work in simplifying -4.5 + 4.2 + 5.6 - 7.3. Identify
any errors in his work or in his reasoning. Write feedback to Andrew
explaining the mistake he made and how to correct his work.
Answer:
hence the answer for given problem is -2, but andrew get -21.6 which is wrong
Step-by-step explanation:
George is building a fence around a rectangular dog run. He is using his house as one side of the run. The area of the dog run will be 240 square feet. The length of the run is 30 feet, and the width is (30 minus x) feet. The diagram below shows his plan.
Recall the formulas for area and perimeter: A = lw and P = 2l + 2w.
A rectangle labeled Dog run has a length of 30 feet and width of (30 minus x) feet.
How many feet of fencing will George need for the dog run?
The length of the fencing needed for the dog run will be 76 feet
What is the length?
The measure of the size of any object or the distance between the two endpoints will be termed the length.
Given that:-
Area of Dog Run = 240
Length of Dog Run = 30
The solution will be done as below:-
Width of Dog Run = 30 - x
Area of Dog Run: 30 * (30 - x) = 240
240 / 30 = 30 - x
8 = 30 - x
x = 24
Width of Dog Run = 8
How many feet of fencing will George need for the dog run?
Perimeter of Dog Run = 2Length + 2Width
Perimeter of Dog Run = (2 * 30) + (2 * 8)
Perimeter of Dog Run = 60 + 16
The perimeter of Dog Run = 76
Therefore the length of the fencing needed for the dog run will be 76 feet
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Answer:
C. 76 feet
Step-by-step explanation:
windows
The article What is i* if I =o
questions:you have been offered a scholarships to further your studies in Australia
write a diary entry sharing your experience about it
Just need some help with this question.
A northbound train left at noon. Two hours later, a southbound train left the same station. At five o'clock the two trains were 210 miles apart. Find the rate of each train of the northbound train traveled 10 miles per hour faster than the southbound train.
The rate of each train of the northbound train traveled 10 miles per hour faster than the southbound train 26 miles per hour.
what is speed?
Speed is the time rate at which an object is moving along a path
Let the speed of southbound train is x miles/hour
so, northbound train has speed of x+10 miles/hour.
then, Speed of both train = Total distance/ total time
(x+10+x)=210/5
2x+10= 210/5
2x+10= 42
2x= 32
x= 16 miles per hour (southbound train)
northbound train: 16+10=26 miles per hour
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can you please help me? 2.1.3
[tex]~~(2I_2)A\\\\\\=2\begin{bmatrix} 1&0\\0&1 \end{bmatrix}\begin{bmatrix} 4&2\\3&-3 \end{bmatrix}\\\\\\=\begin{bmatrix} 2&0\\0&2 \end{bmatrix}\begin{bmatrix} 4&2\\3&-3 \end{bmatrix}\\\\\\=\begin{bmatrix} 8+0&4+0\\0+6&0-6 \end{bmatrix}\\\\\\=\begin{bmatrix} 8&4\\6&-6 \end{bmatrix}[/tex]
Drag the tiles to the correct boxes to complete the pairs.
Match each radical equation with the corresponding value of x.
5√x^4=81
3√x^4= 625
3√x^5=32
4√x^3=64
The value of x for each radical equation is 243 , 125,8,256 respectively.
The correct question is attached as an image with the answer.
What is a Radical Equation ?
When in an equation the variable is under a radical it is called a Radical Equation.
It is given in the question that
which radical matches to the solution
[tex]\rm \sqrt[5]{x^4} = 81 \\\\\sqrt[3]{x^4} = 625\\\\\sqrt[3]{x^5} = 32\\\\ \sqrt[4]{x^3}=64[/tex]
[tex]\rm x^4 = 81^5\\x = 243[/tex]
[tex]\rm x^4 = 625^3\\x = 125\\[/tex]
[tex]x^5 = 32^3\\x = 8\\[/tex]
[tex]\rm x ^3 = 64^4\\x = 256[/tex]
Therefore for each radical the value of x is determined.
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ow is the product of 3 and –2 shown using integer tiles?
3 positive tiles and 2 negative tiles.
3 negative tiles and 2 positive tiles.
3 sets of 2 negative tiles.
3 sets of 2 positive tiles.
The correct answer is option C which is 3 sets of 2 negative tiles.
What is multiplication?Multiplication is the process of determining the product of two or more numbers in mathematics.
We have the product of 3 and -2. We can represent this as 3 is equivalent to -2 negative tiles.
When we add -2 three times -2 - 2 -2 = -6 we will get -6 that is 3 sets of 2 negative tiles.
So when we multiply it by 3, we end with 3 sets of 2 negative tiles. This is because the order of the factors does not change the product.
Therefore the correct answer is option C which is 3 sets of 2 negative tiles.
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Which of the following describes the function shown in the table below? x y -4 16 -1 2 2 0.25 4 0.0625 5 0.03125 exponential, there is a continual rate of growth exponential, there is a continual rate of decay or decrease quadratic, there is a second degree change in the y-values quadratic, there is a constant difference between consecutive y-values
A statement which best describes the function shown in the table is: B. exponential, there is a continual rate of decay or decrease.
What is an exponential function?An exponential function is a mathematical function whose numerical values (numerals) are generated by a constant that is raised to the power of an argument.
Given the following table:
X : -4 -1 2 4 5
Y : 16 2 0.25 0.0625 0.03125
Next, we would calculate the rate of decay for the y-values:
16/2 = 2/0.25 = 0.25/0.0625 = 0.0625/0.03125 = 8
Rule: The y-values constantly decreases by 1/4 or 0.25 of the previous number.
In conclusion, we can logically deduce that the function shown in the table is exponential, there is a continual rate of decay or decrease.
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Answer: It's B
Step-by-step explanation:
I got it right
Write the following expression without negative exponents and without parentheses.
(9x)^-1
Answer:
1/9x
Step-by-step explanation:
The parenthesis around the 9x means that the -1 exponent goes to the whole thing (the 9x) To "fix" a negative exponent push the 9x across a fraction bar to make the exponent positive. When you push the 9x down across a fraction bar that leaves a 1 on top of the fraction, that has nothing to do with the 1 exponent. Its a 1 on top because when you push the 9x down there is nothing left on top so it is a 1.
(9x)^-1
= 1/9x
To be clear, this is a 1 one top and a 9x on the bottom of a fraction.
Which phrase best describes this polygon?
concave octagon
concave nonagon
convex nonagon
convex octagon
Answer:
Concave octagon
Step-by-step explanation:
Concave octagon:
This polygon has 8 sides. So it is a octagon. If one interior angle measures more than 180, then it is concave polygon.So, this is a concave octagon
Nell is analyzing a quadratic function f(x) and a linear function g(x) will they intersect
Which is the graph of y = [X] - 2?
Answer:
Top right graph
Step-by-step explanation:
Graph is in attached image.
For what value of x will the line passing through the following pairs of points be
vertical?
F(3-2x, 12) and G(x +9, 7)