Answer:
The conjecture is that the sum is [tex]30^2+30=930[/tex].
Step-by-step explanation:
I don't see your table... but let's see if we can make a conjecture about the sum of the first 30 positive even numbers.
What is the sum of the first even number? 2=2
What is the sum of the first two even numbers? 2+4=6
What is the sum of the first three even numbers? 2+4+6=12
What is the sum of the first four even numbers? 2+4+6+8=20
What is the sum of the first five even numbers? 2+4+6+8+10=30
What is the sum of the first six even numbers? 2+4+6+8+10+12=42
Alright, let's stop there for a second.
So we have the following sequence of numbers to find a pattern for:
2,6,12,20,30,42,...
Let's look at the common differences:
6-2 , 12-6 , 20-12 , 30-20, 42-30,...
4 , 6 , 8 , 10 , 12
No common difference here so let's move on too the second common differences:
6-4 , 8-6, 10-8, 12-10
2 , 2 , 2 , 2
So there is a 2nd common difference which means the pattern is a quadratic.
So our expression is of the form [tex]ax^2+bx+c[/tex]
Let's plug in our numbers to come up with a system to solve:
If x=1 , then [tex]ax^2+bx+c=2[/tex]
That is, [tex]a(1)^2+b(1)+c=2[/tex] .
Simplifying this gives: [tex]a+b+c=2[/tex].
If x=2, then [tex]ax^2+bx+c=6[/tex]
That is, [tex]a(2)^2+b(2)+c=6[/tex]
Simplifying this gives: [tex]4a+2b+c=6[/tex].
If x=3, then [tex]ax^2+bx+c=12[/tex]
That is [tex]a(3)^2+b(3)+c=12[/tex]
Simplifying this gives: [tex]9a+3b+c=12[/tex].
So we have this system of equations:
a+ b+ c=2
4a+2b+ c=6
9a+3b+ c=12
I'm going to set this up as a matrix:
[ 1 1 1 2 ]
[4 2 1 6 ]
[9 3 1 12]
Multiply first row by -4:
[ -4 -4 -4 -8 ]
[ 4 2 1 6 ]
[ 9 3 1 12]
Add equation 1 to 2:
[ -4 -4 -4 -8]
[ 0 -2 -3 -2]
[ 9 3 1 12]
Divide first row by -4:
[ 1 1 1 2]
[ 0 -2 -3 -2]
[9 3 1 12]
Multiply top row by -9:
[-9 -9 -9 -18]
[0 -2 -3 -2]
[ 9 3 1 12]
Add equation 3 to 1:
[0 -6 -8 -6]
[0 -2 -3 -2]
[9 3 1 12]
Multiply the second equation by -3:
[ 0 -6 -8 -6]
[0 6 9 6]
[9 3 1 12]
Add equation 1 to 2:
[0 -6 -8 -6]
[0 0 1 0]
[9 3 1 12]
Let's stop there the second row gives us c=0.
So the first row gives us -6b-8c=-6 where c=0 so -6b-8(0)=-6.
Let's solve this:
-6b-8(0)=-6
-6b-0=-6
-6b =-6
b =1
So we have b=1 and c=0 and we haven't used that last equation yet:
9a+3b+c=12
9a+3(1)+0=12
9a+3+0=12
9a+3=12
9a=9
a=1
So your expression for the pattern is [tex]x^2+x+0[/tex] or just [tex]x^2+x[/tex].
Let's test it out for one of our terms in our sequence:
"What is the sum of the first four even numbers? 2+4+6+8=20"
So if we plug in 4 hopefully we get 20.
[tex]4^2+4[/tex]
[tex]16+4[/tex]
[tex]20[/tex]
Looks good!
Now we want to know what happens when you plug in 30.
[tex]30^2+30[/tex]
[tex]900+30[/tex]
[tex]930[/tex]
If you don't like this matrix way, I can think of something else let me.
Factor completely x3 + 4x2 + 8x + 32. (x + 4)(x2 + 8) (x − 4)(x2 − 8) (x − 4)(x2 + 8) (x + 4)(x2 − 8)
[tex]\tt x^3+4x^2+8x+32\\\\=x^2(x+4)+8(x+4)\\\\=\boxed{\tt(x+4)(x^2+8)}[/tex]
Answer:
(x+4)(x^2+8)
Step-by-step explanation:
x^3 + 4x^2 + 8x + 32
We will factor by grouping
Factor out an x^2 from the first two terms and an 8 from the last two terms
x^2 (x+4) + 8(x+4)
Now we can factor out an (x+4)
(x+4)(x^2+8)
Answer all questions thanks
Answer:
a)[tex]40\pi ft^3[/tex]
b)125.68ft³
Step-by-step explanation:
The formula for volume of a cylinder is
[tex]V=\pi *r^2*h[/tex]
Given in the question
Diameter=4ft and height =10ft
You know radius is half the diameter,
radius, r=4/2= 2ft
Finding the volume in terms of pi
[tex]V=\pi *r^2*h\\\\V=\pi *2*2*10\\\\\\V=40\pi ft^3[/tex]
Substitute pi with its value,
pi=3.142
Find volume
[tex]V=\pi *r^2*h\\\\\\V=3.142*2*2*10\\\\V=125.68ft^3[/tex]
The following table shows a portion of a three-year amortization schedule.
Use the information in the table to decide which of the following statements is true.
a.
The payment amount changes each month.
b.
The amount applied to the principal is decreasing each month.
c.
The amount applied to the principal is increasing each month.
d.
The amount applied to interest is increasing each month.
Answer:
Option C is correct.
Step-by-step explanation:
If we look at the table,
The amount of Principal of Month 13 is $325.82
The amount of Principal of Month 14 is $328.19
The amount of Principal of Month 15 is $330.57
So, the amount of principal is increasing every month.
Rest of the options seems incorrect.
So, Option C The amount applied to the principal is increasing each month.
is correct.
Answer:
The answer is C
Step-by-step explanation:
I got the right answer on the test in e2020
Which numbers are real numbers ?
Select ALL correct answers.
Answer:
A, e, F
Step-by-step explanation:
the first term of a geometeic series is 1, the common ratio is 2, and the sum of the series is 63. How many terms are there in this geonetric series?
Answer:
6 terms
Step-by-step explanation:
To find the number of terms we will use the formula:
Sn = a(1-r^n)/1-r
where a is the first term= 1
r is the common ratio = 2
Sn is the sum of series = 63
n is the number of terms = ?
Now substitute the values in the formula:
Sn = a(1-r^n)/1-r
63= 1(1-2^n)/1-2
63=1(1-2^n)/-1
63*-1 = 1(1-2^n)
-63 = 1-2^n
Now combine the constants:
-63-1= -2^n
-64= -2^n
64= 2^n
2^6 = 2^n
When the base are same powers are equal to each other.
6=n
Thus there are 6 terms....
Answer:
Please mark as Brainliest :)
Step-by-step explanation:
A box can hold 24 crayons. How many boxes must be used to put all of Mr.Syed's 98 crayons into boxes?
Answer:
5 boxes
Step-by-step explanation:
Given:
Total crayons : 98
Capacity of each box : 24
Number of boxes of 24 crayons needed to hold 98 crayons,
= 98 ÷ 24
= 4.08 boxes.
However since we cannot have 0.08 of a box, we must round this up to the next larger whole number of boxes.
4.08 rounded up to next whole number = 5
Hence he will need 5 boxes to hold all 98 crayons
write three equivalent fractions for 60.1% with denominators of 1,000, 100, and 10
Final answer:
Three equivalent fractions for 60.1% with different denominators are found by converting the percent to a decimal and then to the requested fractions. The fractions are 601/1,000 (simplest form), 60.1/100 simplifies to 601/1,000, and 6.01/10 simplifies to 601/1,000.
Explanation:
To write three equivalent fractions for 60.1% with denominators of 1,000, 100, and 10, we first convert the percent to a decimal by dividing by 100. So, 60.1% becomes 0.601. This decimal can then be written as a fraction with the various denominators.
With a denominator of 1,000: 0.601 × 1,000/1,000 = 601/1,000
With a denominator of 100: 0.601 × 100/100 = 60.1/100 which simplifies to 601/1,000
With a denominator of 10: 0.601 × 10/10 = 6.01/10 which simplifies to 601/1,000
Note that 601/1,000 is the simplest form for the fraction equivalent of 60.1%. When we adjust for the different denominators, we always arrive back at the equivalent fraction of 601/1,000 because it is the simplest form.
Evaluate x^2+3.5when x=1/2
Answer:
3.75.
Step-by-step explanation:
Substitute x = 1/2 into x^2 + 3.5:
(1/2)^2 + 3.5
= 1/4 + 3.5
= 3.75.
Answer:
3.75Step-by-step explanation:
[tex]\text{Put the value of}\ x=\dfrac{1}{2}=0.5\ \text{to the expression}\ x^2+3.5:\\\\0.5^2+3.5=0.25+3.5=3.75[/tex]
Solve the following literal equation for x then for y
8x+5y=10
Answer:
see explanation
Step-by-step explanation:
Given
8x + 5y = 10 ← solve for x by subtracting 5y from both sides
8x = 10 - 5y ( divide both sides by 8 )
x = [tex]\frac{10-5y}{8}[/tex]
------------------------------------------------------------
Solve for y by subtracting 8x from both sides
5y = 10 - 8x ( divide both sides by 5 )
y = [tex]\frac{10-8x}{5}[/tex]
What equation is the inverse of y=x^2+16
Answer:
[tex]y = \sqrt{x - 16} [/tex]
Step-by-step explanation:
[tex]y = {x}^{2} + 16[/tex]
[tex]x = {y}^{2} + 16[/tex]
[tex]x - 16 = {y}^{2} [/tex]
[tex] \sqrt{x - 16} = y[/tex]
Answer:
f^-1 (x) = ±sqrt(x-16)
Step-by-step explanation:
To find the inverse of a function, exchange x and y and solve for y
y=x^2+16
Exchange x and y
x = y^2 +16
Then solve for y
Subtract 16 from each side
x-16 = y^2 +16-16
x-16 = y^2
Take the square root of each side
±sqrt(x-16) = sqrt(y^2)
±sqrt(x-16) = y
The inverse
f^-1 (x) = ±sqrt(x-16)
What is the volume of a cone that has a base with a radius of 4 inches and a height of 12 inches?
Answer: [tex]201.06\ in^3[/tex]
Step-by-step explanation:
The volume of a cone can be calculated with this formula:
[tex]V=\frac{1}{3}\pi r^2h[/tex]
Where "r" is the radius and "h" is the height.
In this case you know that this cone has a base with a radius of 4 inches and a height of 12 inches. Then:
[tex]r=4\ in\\h=12\ in[/tex]
So, in order to calculate its volume, you must substitute these values into the formula. Then:
[tex]V=\frac{1}{3}\pi (4\ in)^2(12\ in)\\\\V=201.06\ in^3[/tex]
Answer:
The volume of given cone = 200.96 inches³
Step-by-step explanation:
Points to remember
Volume of cone =(1/3) πr²h
Where 'r' is the radius and 'h' is the height of the cone
To find the volume of cone
Here r = 4 inches and h = 12 inches
Volume = (1/3)πr²h
= (1/3) * 3.14 * 4² *12
= 200.96 inches²
Therefore the volume of given cone = 200.96 inches³
I need help with this
Answer:
Step-by-step explanation:
Setting the denominator = to 0, we get x = -4. This is the location of the discontinuity.
Setting the numerator = to 0 and solving for x yields the zeros: They are {-2, -4}.
Event v occurs 28% of the time on Tuesdays.
and event v and event w occur together 19%
of the time on Tuesdays. Given that event v
occurs on a Tuesday, what is the probability
that event w occurs with event v?
A. 47%
B. 58%
C. 68%
D. 75%
Answer:
68%
Step-by-step explanation:
Probability of occurrence of Event v = P(v) = 28% = 0.28
Probability of occurrence of both Events v and Event w together = P(v and w) = 19% = 0.19
We have to find what is the probability that event w occurs with event v given that event v occurs on a Tuesday. This is a conditional probability. In other words we have to find what is the probability of event w given that event v occurs of Tuesday. i.e we have to find P(w|v)
The formula to calculate this conditional probability is:
[tex]P(w|v) = \frac{P(v \cap w)}{P(v)}[/tex]
Using the given values, we get:
[tex]P(w|v) = \frac{0.19}{0.28}\\\\ P(w|v) = 0.68\\\\ P(w|v) = 68\%[/tex]
Therefore, the probability that even w will occur with event v given that event v occurs on Tuesday is 68%
How many x-intercepts does the graph of y = 2x2 + 5x + 9 have?
Answer:
The graph of y = 2x² + 5x + 9 has zero x-intercepts.Step-by-step explanation:
x-intercepts are for y = 0.
Put y = 0 to the equation y = 2x² + 5x + 9:
2x² + 5x + 9 = 0
Calculate the discriminant of quadratic equation ax² + bx + c = 0:
Δ = b² - 4ac
if Δ < 0, then an equation has no solution
if Δ = 0, then an equation has one solution
if Δ > 0, then an equation has two solution.
2x² + 5x + 9 = 0
a = 2, b = 5, c = 9
Δ = 5² - 4(2)(9) = 25 - 72 = -47 < 0 - no solution
There are zero x- intercepts of given function.
X- Intercepts:To find x- intercepts, substitute [tex]y=0[/tex] in the function.
Given function is, [tex]y=2x^{2} +5x+9[/tex]
Substitute [tex]y=0[/tex] in above equation.
[tex]2x^{2} +5x+9=0\\\\[/tex]
The graph of function where crosses the x- axis that point is known as x- intercepts of function.
Graph of given function attached below, It is observed that graph will not crosses the x axis at any point.
Thus, There are zero x- intercepts of given function.
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Complete the table for the given rule. Rule: y=X/4(fraction)
Table
X Y
4
2
9
What would X be i am really confused here does the fraction mean divide the numbers or something else
The complete table of values for the equation y = x/4 is
x 4 2 9
y 1 1/2 9/4
Completing the table of values for the equationFrom the question, we have the following parameters that can be used in our computation:
y = x/4
The missing y values are at x = 4, 2 and x = 9
Substitute the known values in the above equation, so, we have the following representation
y = 1/4 * 4 = 1
y = 1/4 * 2 = 1/2
y = 1/4 * 9 = 9/4
So, the missing values are 1, 1/2 and 9/4
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Final answer:
The rule 'y = X/4' describes division of X by 4 to find Y. The student completes the table by dividing each X value by 4 to find the corresponding Y values, providing an organized data set.
Explanation:
The student is asking about the operation indicated by the rule y = X/4, which involves division. The fraction bar signifies division. Given this rule, we can complete the table for y when x-values are provided.
For x = 4: y = 4/4 = 1
For x = 2: y = 2/4 = 0.5
For x = 9: y = 9/4 = 2.25
To organize the data in the table, you simply divide the x-value by 4 to find the corresponding y-value.
On Monday there were 92 fish in the
fish tank. On Tuesday 4 fish were
sold, on Wednesday 1 fish was sold,
on Thursday 4 fish were added to the
tank, and on Friday 3 fish were sold.
How many fish were in the fish tank
at the end of the week?
Answer:
35
Step-by-step explanation:
92-5+4-3=35
Which expression is equivalent to (5x^5)(4x)^3
Answer:
5x. 5x. 5x. 5x. 5x. 4x. 4x. 4x
Step-by-step explanation:
Just seperate all of the numbers out. Because 5x is raised to the fifth it is being multiplied by itself 5 times, the same goes for 4.
Which of the following geometric objects occupy one dimension?
Check all that apply
A.Point
B.Segment
C.Line
D.Triangle
E.Plane
F. Ray
Answer:
Segment, line and Ray
Step-by-step explanation:
Lets discuss all the options one by one.
Point:
A point has no length, no width and no depth. It is 0 dimensional.
Line:
A line extends in both directions. It is straight and has no thickness. It is 1 dimensional
Segment:
It is a part of a line bounded by two distinct end points. As it is a part of a line than it is also 1 dimensional.
Triangle:
A triangle has three sides and three corners. It is two dimensional figure. Its interior or oval is also two dimensional.
Plane:
A plane is a two dimensional flat surface with no thickness.
Ray:
A ray is a part of a line with single end point. It goes off in a particular direction infinitely. As it is a part of a line hence it is also 1 dimensional:
Thus according to the above description Segment, Line and Ray are one dimensional
Answer:
C. Line. F. Ray B. Segment ( but see below.)
Step-by-step explanation:
A line is one dimensional, so is a Ray.
As for a segment it depends on what you mean . A line segment is one dimensional, but a segment of a circle has 2 dimensions.
Recall the definition of a polynomial expression. Find two polynomial expressions whose quotient, when simplified, is 1/x. Use that division problem to determine whether polynomials are closed under division. Then describe how the other three operations—addition, subtraction, and multiplication—are different from division of polynomials.
Please help, thanks so much!!
Answer:
Step-by-step explanation:
A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Actually 1 and x both are polynomials. Therefore 1/x is already a quotient of polynomials.
Two polynomial expressions whose quotient, when simplified, is 1/x
x+1/x^2+x
Take the common of the denominator
x+1/x(x+1)
x+1 will be cancelled out by x+1
1/x
1/x is not a polynomial, so polynomials are not closed under division.
Polynomials are closed under addition, subtraction and multiplication. Addition and multiplication are associative and commutative.
There is an additive identity which is 0.
There is a multiplicative identity which is 1....
Polynomial expressions involve variables, coefficients, and exponents, combined by addition, subtraction, and multiplication. They are not closed under division, as shown by the quotient 1/x, which is not a polynomial. However, polynomials are closed under addition, subtraction, and multiplication.
Explanation:A polynomial expression is a mathematical expression that can involve variables, coefficients, and exponents, that are combined using addition, subtraction, and multiplication operations. For your question, we can consider two polynomial expressions: x and 1. The quotient of these two expressions is 1/x.
However, 1/x is not a polynomial expression, because it involves division by a variable. This shows that polynomial expressions are not closed under division.
In contrast, polynomials are closed under the operations of addition, subtraction, and multiplication. For example, if we add, subtract or multiply any two polynomials we will always get another polynomial.
These operations do not generate any new type of expressions outside the realm of polynomials, unlike division, which can produce non-polynomial results, such as rational expressions.
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Evaluate the following expression when x = 5 and z = 3.
z
-
3x=
Answer:-12
Step-by-step explanation:
Z-3x=
=3-3(5)
=3-15
=-12
Answer:
-12
Step-by-step explanation:
3 - 3(5)
3 -15 = -12
the answer to the question
Liliana wants to start a seventh-grade computer club at Hamden Middle School. She surveyed 20 seventh-grade students at the town park. She asked each student how many hours they spend on their computers each week. She obtained the following results.
8, 15, 0, 11, 12, 13, 16, 13, 0, 4, 17, 14, 30, 13, 5, 12, 1, 13, 12, 21
What is the ratio of the total number of students who used their computers to the total number of students surveyed?
2/18 or 1/9
18/2 or 9/1
2/20 or 1/10
18/20 or 9/10
Answer:
18/20 or 9/10
Step-by-step explanation:
The list shows the amount of hours each student USED the computer.
Wherever we see a "0", that means he/she DID NOT USE computer.
There are 2 "zeroes", so 2 didnt use out of 20, so number of student who DID USE is 20 - 2 = 18
To find ratio of total number of students who used to the total number surveyed, it is going to be:
18/20 or 9/10
Answer:
Ratio would be 9 : 10
Step-by-step explanation:
Liliana surveyed 20 students that how many hours they spend on their computers each week.
The results are as follows
8, 15, 0, 11, 12, 13, 16, 13, 0, 4, 17, 14, 30, 13, 5, 12, 1, 13, 12, 21
In the above results 2 students did not use computers out of 20 students.
Therefore, The total number of students who used their computers = 20 - 2 = 18
So the ratio would be 18 : 20 or 9 : 10
Complete the table for the given rule y=6x-4
Answer:
1 = 2
3 = 14
10 = 56
Step-by-step explanation:
y = 6x - 4
y = 6 - 4
y = 18 - 4
y = 60 - 4
Step-by-step explanation:
In first there is 2
ln second there is 14
ln third there is 56
I hope this is right answer!
Solve (x + 2<5) n (x-7>-6).
{x11
{x1x < 3 or x> 1)
{all real numbers)
For this case we must solve [tex]x + 2 <5[/tex]intersected with [tex]x-7> -6[/tex]
So, we have:
[tex]x + 2 <5\\x <5-2\\x <3[/tex]
The solutions are given by all strict minor numbers to 3.
[tex]x-7> -6\\x> -6 + 7\\x> 1[/tex]
The solutions are given by all the major strict numbers to 1.
If we intersect the solutions of both equations we have [tex]x <3[/tex] ∩ [tex]x> 1[/tex].
The intersection is given by [tex]1 <x <3[/tex]
Answer:
The solution interval is (1,3)
solve 2x^2 + 8 = 0 by graphing the related function
The solutions to the equation [tex]\( 2x^2 + 8 = 0 \)[/tex] are complex numbers [tex]\( x = 2i \) and \( x = -2i \), where \( i \)[/tex] is the imaginary unit.
The statement that the graph of [tex]\( 2x^2 + 8 = 0 \)[/tex] does not cross the x-axis is correct, as the graph of this equation represents a parabola that opens upwards and is centered above the x-axis, meaning it has no real roots.
To find the solutions for the equation [tex]\( 2x^2 + 8 = 0 \)[/tex], we can start by isolating [tex]\( x^2 \):[/tex]
[tex]\[ 2x^2 = -8 \][/tex]
Dividing both sides by 2:
[tex]\[ x^2 = -4 \][/tex]
Now, to solve for ( x ), we take the square root of both sides:
[tex]\[ x = \pm \sqrt{-4} \][/tex]
[tex]\[ x = \pm 2i \][/tex]
So, the solutions to the equation [tex]\( 2x^2 + 8 = 0 \)[/tex] are complex numbers [tex]\( x = 2i \) and \( x = -2i \), where \( i \)[/tex] is the imaginary unit.
These are the roots of the equation, even though they do not lie on the real number line.
Hence, while there are no real solutions, there are indeed complex solutions for the given function.
PLEASE
The three sides of a triangle measure 6, 15, and n. What is the range of possible lengths of n?
6 < n < 15
6 < n < 21
9 < n < 21
9 < n < 15
Answer:
9 < n < 21
Step-by-step explanation:
15-6 < n < 15+6
9 < n<21
We are limited by the other two sides of the triangle
It can be no smaller than the other two sides subtracted and no larger than the other two sides added
If AB = 8, BC = 5, and CA = 7. list the angles of
AABC in order from smallest to largest measure.
Answer:
∠A, ∠B, ∠C
Step-by-step explanation:
In a triangle the smallest angle is opposite the shortest side, and the largest angle is opposite the longest side, therefore:
∠A < ∠B < ∠C
Answer:
a,b,c
Step-by-step explanation:
The angle opposite the longest side has the largest angle measure, the angle opposite the shortest side has the most small angle measure etc.
Since BC is the shortest line, the angle opposite will have the smallest measure, this is ∠a.
CA is in the middle with length this means that the angle opposite of it will have an angle measure in between that of the other angle measures. This is ∠b
AB has the longest length, therefore, the angle opposite it will have the largest measure, this is ∠c
The student council is selling sweatshirts with the school name and enim...
shirts come in three different colors, two different styles and three different sizes.
How many different types of sweatshirts are available?
In order to find all the types of sweatshirts, we simply have to multiply all the numbers by all the factors.
3 colors * 2 different styles * 3 different sizes = 18 different types of sweatshirts
The total of 18 different types of sweatshirts are available if shirts come in three different colours, two different styles and three different sizes.
What is permutation and combination?A permutation is the number of different ways a set can be organized; order matters in permutations, but not in combinations.
We have:
The student council is selling sweatshirts with the school name and shirts come in three different colours, two different styles and three different sizes.
Total number of different sweatshirts = 3×2×3
= 18
Thus, the total of 18 different types of sweatshirts are available if shirts come in three different colours, two different styles and three different sizes.
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A circle has a circumference of 32.4 mm. Find the radius and the diameter. Round your answers to two decimal places.
Answer:
r=5.15
d=10.31
Step-by-step explanation:
Given:
circumference of circle=32.4 mm
radius=?
diameter=?
formula of circumference of circle is given as:
C=2πr
Putting value we get
32.4=2πr
r=32.4/2π
r=5.15
d=2r
d=2(5.15)
d=10.31 !
For this case we have that by definition, the circumference of a circle is given by:
[tex]C = \pi * d[/tex]
Where:
d: It is the diameter of the circle
We have as data that:
[tex]C = 32.4 \ mm[/tex]
Taking[tex]\pi = 3.14[/tex]
We have:
[tex]32.4 = 3.14 * d\\d = \frac {32.4} {3.14}\\d = 10.32 \ mm[/tex]
The radius is given by:
[tex]\frac {10.32} {2} = 5.16 \ mm[/tex]
Answer:
[tex]Diameter: 10.32 \ mm\\Radius: 5.16 \ mm[/tex]
Which of the following theorems verifies that ABC=SPR?
LOOK AT PICTURE
Answer:
B. is correct
Answer: B. HA
Step-by-step explanation:
The HA (Hypotenuse-Angle) theorem says that if the hypotenuse and an acute angle of two right triangles are congruent, then those triangles are congruent.In the given picture we have two right triangles ΔABC and ΔSPR.
In ΔABC and ΔSPR, we have
AB≅SP [Hypotenuse]
∠B≅∠P [Angle]
Thus by HA theorem we have
ΔABC ≅ ΔSPR
What is the first step in solving In(x - 1) = In6 - Inx for x?
Answer:
The first step would to be use quotient rule.
3
Step-by-step explanation:
ln(x-1)=ln(6)-ln(x)
The first step would to be use quotient rule there on the right hand side:
ln(x-1)=ln(6/x)
*Quotient rule says ln(a/b)=ln(a)-ln(b).
Now that since we have ln(c)=ln(d) then c must equal d, that is c=d.
ln(x-1)=ln(6/x)
implies
x-1=6/x
So you want to shove a 1 underneath the (x-1) and just cross multiply that might be easier.
[tex]\frac{x-1}{1}=\frac{6}{x}[/tex]
Cross multiplying:
[tex]x(x-1)=1(6)[/tex]
Multiplying/distribute[/tex]
[tex]x^2-x=6[/tex]
Subtract 6 on both sides:
[tex]x^2-x-6=0[/tex]
Now this is not too bad to factor since the coefficient of x^2 is 1. All you have to do is find two numbers that multiply to be -6 and add up to be -1.
These numbers are -3 and 2 since -3(2)=-6 and -3+2=-1.
So the factored form of our equation is
[tex](x-3)(x+2)=0[/tex]
This implies that x-3=0 or x+2=0.
So solving x-3=0 gives us x=3 (just added 3 on both sides).
So solve x+2=0 gives us x=-2 (just subtracted 2 on both sides).
We need to see if these are actually the solutions by plugging them in.
Just a heads up: You can't do log(negative number).
Checking x=3:
ln(3-1)=ln(6)-ln(3)
ln(2)=ln(6/3)
ln(2)=ln(2)
This is true.
Checking x=-2:
ln(-2-1)=ln(6)-ln(-2)
ln(-3)=ln(6)-ln(-2)
We don't need to go further -2 makes the inside of our logarithms negative above.
The only solution is 3.
Step-by-step explanation:
The first step is to create the domain of this equation.[tex]\ln(x-1)=\ln6-\ln x\\\\D:\ x-1>0\ \wedge\ x>0\\\\x>1\ \wedge\ x>0\Rightarrow x>1[/tex]
[tex]\ln(x-1)=\ln6-\ln x\qquad\text{use}\ \log_ab-\log_ac=\log_a\dfrac{b}{c}\\\\\ln(x-1)=\ln\dfrac{6}{x}\iff x-1=\dfrac{6}{x}\\\\\dfrac{x-1}{1}=\dfrac{6}{x}\qquad\text{cross multiply}\\\\x(x-1)=(1)(6)\qquad\text{use the distributive property}\ a(b+c)=ab+ac\\\\(x)(x)+(x)(-1)=6\\\\x^2-x=6\qquad\text{subtract 6 from both sides}\\\\x^2-x-6=0\\\\x^2+2x-3x-6=0\\\\x(x+2)-3(x+2)=0\\\\(x+2)(x-3)=0\iff x+2=0\ \vee\ x-3=0\\\\x+2=0\qquad\text{subtract 2 from both sides}\\x=-2 \notin D\\\\x-3=0\qquad\text{add 3 to both sides}\\x=3\in D[/tex]
Solution:
x = 3