Explanation
Factorial, in mathematics, the product of all positive integers less than or equal to a given positive integer and denoted by that integer and an exclamation point.
[tex]a![/tex]so, to evaluate the expression we need to apply the definition
hence
[tex]\begin{gathered} 12\text{ ! = 12}\cdot11\cdot10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1 \\ 12\text{ ! =}479001600 \end{gathered}[/tex]I hope this helps you
Find the surface area and volume of the sphere. Round your answers to the nearest whole number.C = 4 in.The surface area is aboutsquare inchesThe volume is aboutcubic inches
The circumference of a sphere with radius r is given by the following formula:
[tex]C=2\pi r[/tex]Isolate r from the equation and substitute the value of C to find the raius of the sphere:
[tex]\begin{gathered} \Rightarrow r=\frac{C}{2\pi} \\ \Rightarrow r=\frac{4\pi\text{ in}}{2\pi} \\ \Rightarrow r=2\text{ in} \end{gathered}[/tex]The surface area of a sphere with radius r is given by:
[tex]S=4\pi r^2[/tex]The volume of a sphere with radius r is given by:
[tex]V=\frac{4}{3}\pi r^3[/tex]Substitute r=2 in on each formula to find the volume and surface area of the sphere:
[tex]\begin{gathered} S=4\pi(2in)^2 \\ =4\pi\cdot4in^2 \\ =16\pi in^2 \\ =50.26\ldots in^2 \\ \approx50in^2 \end{gathered}[/tex][tex]undefined[/tex]You have a rectangular backyard that is 90 feet wide. It has an area of 10,800 square feet. You are putting a fence along one length of the yard.Use the formula for the area of a rectangle A = l * w, where I is the length, and w is the width. Find the length of your backyard.
ANSWER:
120 feet
STEP-BY-STEP EXPLANATION:
We hve the following:
A = 10800 ft²
w = 90 ft
The area of a rectangle is the product between the length and the width, we do not know the length but we do know the area, therefore, we calculate the length as follows:
[tex]\begin{gathered} A=w\cdot l \\ \\ \text{ We replacing} \\ \\ 10800=90\cdot l \\ \\ l=\frac{10800}{90} \\ \\ l=120\text{ ft} \end{gathered}[/tex]The length of your backyard is equal to 120 feet
how can I find which statements can be deducted from the picture
The following statements can be deduced by knowing some previous knowledge from Parallelism.
2) Examining and commenting
∠1 ≅ ∠2 FALSE This is a linear pair. So they are supplementary angles, not congruent ones.
∠5≅ ∠7 TRUE Vertical Angle Theorem states that they are congruent ones.
m∠7≅ m∠4 True Linear pair a+ b are supplementary as well.
∠1 ≅ ∠7 True Corresponding Angles are always congruent.
≅
Kennedy goes to a store an buys an item that costs xx dollars. She has a coupon for 35% off, and then a 7% tax is added to the discounted price. Write an expression in terms of xx that represents the total amount that Kennedy paid at the register.
We are given that an item has a cost xx.
First, we will calculate the 35% discount on the total price. To do that we will subtract 35% of the initial cost from the initial cost
To calculate the 35% we multiply the price by 35/100, like this:
[tex]xx\times\frac{35}{100}[/tex]Now we subtract this from the initial price, which was "xx". Subtracting we get:
[tex]P_d=xx-xx\times\frac{35}{100}[/tex]This is the cost with the discount. Now, we will add to this the tax of 7%. First, we calculate the 7% of the price with the discount by multiplying it by 7/100, like this:
[tex](xx-xx\times\frac{35}{100})\times\frac{7}{100}[/tex]Now, we add this to the price with the discount, like this:
[tex]T=(xx-xx\times\frac{35}{100})+(xx-xx\times\frac{35}{100})\times\frac{7}{100}[/tex]Now, we can simplify. We start by using the distributive property on the second parenthesis:
[tex]T=xx-xx\times\frac{35}{100}+xx\times\frac{7}{100}-xx\times\frac{35}{100}\times\frac{7}{100}[/tex]Now we solve the product of 35/100 by 7/100, we get:
[tex]T=T=xx-xx\times\frac{35}{100}+xx\times\frac{7}{100}-xx\times\frac{49}{2000}[/tex]Now we take xx as a co
If {an) is an arithmetic sequence where a1=-23 and the common difference is 6, find a79
Given:
The first term
[tex]a_1=-23[/tex]The common difference, d=6
To find
[tex]a_{79}[/tex]Using the nth term formula,
[tex]\begin{gathered} a_n=a+(n-1)d \\ a_{79}=-23+(79-1)6 \\ =-23+(78)6 \\ =-23+468 \\ =445 \end{gathered}[/tex]Hence, the answer is,
[tex]a_n=445[/tex]I don't understand. I get an answer that doesn't exist. The question is to multiply in a+bi form. (5/2 + 2i)(1/4 - 6i)
Given the below
[tex](\frac{5}{2}+2i)(\frac{1}{4}-6i)[/tex]Multiplication of the vectors in the a+bi form gives
Applying the complex arithmetic rule below
[tex](a+bi)(c+di)=(ac-bd)(ad+bc)i[/tex]The expansion of the vectors gives
[tex](\frac{5}{2}+2i)(\frac{1}{4}-6i)=\frac{5}{2}(\frac{1}{4}-6i)+2i(\frac{1}{4}-6i)[/tex]Opening the brackets
[tex]\begin{gathered} (\frac{5}{2}+2i)(\frac{1}{4}-6i)=\frac{5}{2}(\frac{1}{4}-6i)+2i(\frac{1}{4}-6i) \\ =\frac{5}{8}-\frac{30}{2}i+\frac{2}{4}i-12i^2 \\ \text{Where i}^2=-1 \\ =\frac{5}{8}-\frac{30}{2}i+\frac{2}{4}i-12(-1) \\ ==\frac{5}{8}+12-\frac{30}{2}i+\frac{2}{4}i \end{gathered}[/tex]Simplifying the above expression
[tex]\begin{gathered} =\frac{5+96}{8}+\frac{-60i+2i}{4}=\frac{101}{8}+\frac{-58}{4}i \\ =\frac{101}{8}+\frac{-29}{2}i=\frac{101}{8}-\frac{29i}{2} \\ (\frac{5}{2}+2i)(\frac{1}{4}-6i)=\frac{101}{8}-\frac{29i}{2} \end{gathered}[/tex]Hence, the answer is
[tex]=\frac{101}{8}-\frac{29i}{2}[/tex]Given the graph given I need help with questions A - D
Using the graph and the table we can infer that the value of the premium for the insurance amount of $50,000 is $28.29 .
From the given table we can see that the function f(x) represents the insurance amount and the premium for the male population.
therefore we can simply substitute the values from the table.
a)f(50000) = $ 28.29
f(25,000) = $ 14.15
b)From the given table we can see that the function g(x) represents the insurance amount for the female population.
g(75000) = $ 19.25
g(25000) = $ 6.42
c) at f(x) = 14.15 the value of x is $25000
d) From the graph let us compare each values for f(x) and g(x).
f(20000)>f(20000)
f(25000)>g(25000)
f(50000)>g(50000)
f(75000)>g(75000)
f(100000)>g(100000)
One party will promise another party reimbursement in the event with a specific loss, damage, or injury in exchange for a fee in order to protect oneself from financial loss. It is a risk management technique used primarily to guard against the risk of a potential loss.
Hence we can infer that for all values of x , f(x)>g(x).
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Consider the line . 7x-8y=-1Find the equation of the line that is parallel to this line and passes through the point . (-3,-6)Find the equation of the line that is perpendicular to this line and passes through the point . (-3.-6)
we have the line
7x-8y=-1
Find out the slope of the given line
isolate the variable y
8y=7x+1
y=(7/8)x+1/8
the slope is m=7/8
Part a
Find the equation of the line that is parallel to this line and passes through the point . (-3,-6)
Remember that
If two lines are parallel, then their slopes are equal
so
The slope of the parallel line is m=7/8 too
Find out the equation of the line in slope-intercept form
y=mx+b
we have
m=7/8
point (-3,-6)
substitute and solve for b
-6=(7/8)(-3)+b
b=-6+(21/8)
b=-27/8
therefore
The equation is
y=(7/8)x-27/8Part b
Find the equation of the line that is perpendicular to this line and passes through the point . (-3.-6)
Remember that
If two lines are perpendicular, then their slopes are negative reciprocal
so
The slope of the perpendicular line is m=-8/7
Find out the equation of the line in slope-intercept form
y=mx+b
we have
m=-8/7
point (-3,-6)
substitute and solve for b
-6=-(8/7)(-3)+b
b=-6-(24/7)
b=-66/7
therefore
the equation is
y=-(8/7)x-66/7Savings ($)
200
160
120
80
40
O
5 10 15 20 25
Time (weeks)
Find the constant of proportionality and write an equation for the relationship
The constant of proportionality is
The equation for the relationship is
For the given graph related to savings on y-axis and time on x-axis, the constant of proportionality is equal to 4, and the equation representing the relationship between the savings and the time is given by y = 4x.
As given in the question,
From the graph,
y-axis represents the savings in dollars
x-axis represents the time in seconds
Let us consider savings represents by y and time represents by x.
Form the graph we can see that,
When x = 10 ⇒ y = 40
when x = 15 ⇒ y = 60
Graph represents the straight line so it is linear function.
y = 40
⇒ y= 4(10)
⇒ y= 4x
⇒y ∝ x
⇒Constant of proportionality 'k' = 4
Now, for the equation consider two coordinates on the graph,
(x₁ , y₁) = (10, 40)
(x₂ , y₂) = (15, 60)
( y-y₁)/ (x-x₁) = (y₂ -y₁) / (x₂ - x₁)
⇒( y- 40)/(x-10) = (60 -40)/ (15-10)
⇒ y-40 = 4(x-10)
⇒y = 4x
Therefore, for the given graph related to savings on y-axis and time on x-axis, the constant of proportionality is equal to 4, and the equation representing the relationship between the savings and the time is given by y = 4x.
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I will show the answer options later because I can’t add two pictures
We will have the following:
* If Kristin does not decrease the price of her cakes, her projected weekly revenue from cake sales will be $2500.
*If Kristin decreases the price of her cakes, her projected weekly revalue will be $2520.
*Kristin will obtain the same revenue if she sells the cakes for $24 or $21.
Are there any extraneous solutions? If yes, write the solution in the box, if no write "none"). x=
Given:
[tex]\sqrt[]{x+1}+2=4[/tex]x=3 is the only solution for the given equation.
There is no other solutions for the given equation.
So none is the final answer.
Verify algebratically if each function is odd, even, or neither. For question #5 only
Answer:
[tex]\text{ odd}[/tex]Explanation:
Here, we want to check if the given function is even or odd
To do that, we find g(x) and g(-x)
If g(x) equals g(-x), the the function is even. Otherwise, the function is odd
We find the functions as follows:
[tex]\begin{gathered} g(x)=7x^3\text{ - x} \\ g(-x)=7(-x)^3-(-x) \\ g(-x)=-7x^3\text{ + x} \end{gathered}[/tex]Finally:
[tex]\begin{gathered} \text{ since g(x) }\ne\text{ g(-x) } \\ \text{Function g(x) is odd} \end{gathered}[/tex]The Pentagon building in Washington, D.C., is named because it is in the shape ofa regular pentagon. What is the measure of each interior angle?
The formula to find the sum of the interior angles of a polygon is given below,
[tex]\text{Sum of angles of polygon = (n-2)180}^0[/tex]Number, n, of sides of a regular pentagon is 5 i.e n = 5,
To find the sum of angles in a regular pentagon, substitute for n into the formula above,
[tex]\text{Sum of angles of a pentagon=(5-2)180}^0=3\times180^0=540^0[/tex]To find the measure of each angle of the pentagon, the formula is given below
[tex]\begin{gathered} for\text{ each interior angle=}\frac{Sum\text{ of angles}}{n} \\ \text{Where n = 5} \\ \text{For each interior angle=}\frac{540^0}{5}=108^0 \end{gathered}[/tex]Hence, each interior angle is 108°
A random number generator is programmed to produce numbers with a Unif (−7,7) distribution. Find the probability that the absolute value of the generated number is greater than or equal to 1.5.
We are given the following uniform distribution:
The probability that the absolute value of the number is in the following interval:
[tex]\begin{gathered} -7The probability is the area under the curve of the distribution. Therefore, we need to add both areas. The height of the distribution is:[tex]H=\frac{1}{b-a}[/tex]Where:
[tex]\begin{gathered} a=-7 \\ b=7 \end{gathered}[/tex]Substituting we get:
[tex]H=\frac{1}{7-(-7)}=\frac{1}{14}[/tex]Therefore, the areas are:
[tex]P(\lvert x\rvert>1.5)=(-1.5-(-7))(\frac{1}{14})+(7-1.5)(\frac{1}{14})[/tex]Simplifying we get:
[tex]P(\lvert x\rvert>1.5)=2(7-1.5)(\frac{1}{14})[/tex]Solving the operations:
[tex]P(\lvert x\rvert>1.5)=0.7857[/tex]Therefore, the probability is 0.7857 or 78.57%.
Factorise the following quadratic:
e² - 17e + 70
Answer:
Step-by-step explanation:
I think you're supposed to use the quadratic formula Samanth
Know it?
-b ± [tex]\sqrt{b^{2}-4ac }[/tex] / (2a)
so for starters, let me mention, that if 4ac happens to be greater than [tex]b^{2}[/tex] contrary to what all math teacher say, about not being able to taking the square of a negative number, you can, but you just end up with a complex number in the form of A + Bi , where 'i' represents [tex]\sqrt{-1}[/tex] anyway,
for the given equation
A = 1
B = -17
C = 70
{ -(-17) ± [tex]\sqrt{(-17)^{2}-4*1*70 }[/tex] } / (2*1)
{ 17 ± [tex]\sqrt{289-280}[/tex] } / 2
wow, now i'm glad I mentioned about the 4ac being greater :P
it was close, huh
{ 17 ± [tex]\sqrt{9}[/tex] } / 2
{ 17 ± 3 } /2
let's take each case now, the plus and then the minus
{ 17+3 } /2
20 /2
10
now the minus
{17 - 3 } / 2
14 /2
7
now that i've done all that work, I think we could have just done this by inspection :P
(e-7)(e-10)
anyway, hope that helps, ask if you have any questions :)
I got x > 12 for my answer, but when I checked it, it didn't work. Please help me solve and check!
What is the domain of the following function?A) {-6, -2, 1, 5, 7, 8}B) {-2, 7}C) {-6, 1, 5, 8}D) all Real numbers
In this problem, we have that
The domain or input values are the data set {-6, 1, 5, 8}
The range or output values are the data set [-2,7]
therefore
The answer is the option Csolve for x:
7x=6+5 (3x+3)-x
Answer: x= -3
Step-by-step explanation:
7x = 6+ 15x + 15 - x
7x = 21 + 14x
-7x = 21
x= -3
Twelve students in mrs.taylors class want to start band. Seven students each made a drum.The rest of the students made 2 shakers each. How many shakers were made? Use the bar model. Need help have to show work on this whole page.
Solution
What do I need to find?
Number of shakers
what information do I need to use?
12 students
7 made a drum each
The remain students made 2 shakers each
how I will use the information?
We need to find the number of students who made shakers and then multiply by 2
Solve the problem
12-7 = 5 students
5* 2= 10 shakers
Then the final answer would be:
10 shakers in total
Write the sum of the first three terms in the binomial expansion, expressing the result in simplified form.(x – 4y)^7
ANSWER:
[tex](x-4y)^7=x^7-28x^6y+336x^5y^2\ldots[/tex]STEP-BY-STEP EXPLANATION:
We have the following expression:
[tex]\mleft(x-4y\mright)^7[/tex]In this case we can apply the binomial theorem, which is the following:
[tex](a+b)^n=\sum ^n_{i\mathop=0}(\frac{n!}{i!(n-i)!}a^{n-i}\cdot b^i[/tex]we replace and calculate for the first three terms:
[tex]\begin{gathered} 1st=\sum ^7_{i\mathop{=}0}(\frac{7!}{0!(7-0)!}x^{7-0}\cdot(-4y)^0=1\cdot x^7\cdot1=x^7 \\ 2nd=\sum ^7_{i\mathop{=}1}(\frac{7!}{1!(7-1)!}x^{7-1}\cdot(-4y)^1=7\cdot x^6\cdot-4y^1=-28x^6y \\ 3rd=\sum ^7_{i\mathop{=}2}(\frac{7!}{2!(7-2)!}x^{7-2}\cdot(-4y)^2=21\cdot x^5\cdot16y^2=336x^5y^2 \end{gathered}[/tex]Hello can you please tell me if this is right please Simplify expressions by distributing
distribute the fraction into all the terms inside the parentheses
[tex]8\cdot\frac{1}{2}-4g\cdot\frac{1}{2}[/tex]solve the product
[tex]\frac{8}{2}-\frac{4g}{2}[/tex]simplify the fractions
[tex]4-2g[/tex]Is the ordered pair (2, 7) a solution of the function f(x) = x + 5? *
The ordered pair (2, 7) is NOT a solution of the function
f(x) = x + 5
Explanation:If (2, 7) is a solution of the function f(x) = x + 5, then
f(2) = 7
f(2) = 2 + 2 = 4
Since this is not 7, we conclude that the ordered pair is NOT a solution of the function
The Elkhart Athletic Departments sells T-shirts and Hats at a big game to raise money. They sale the T-shirts for $12 and the Hats for $5. At the last football game they sold a total of 32 items and raised $265. How many T-shirts and Hats were sold at the game?
Let x represent the number of T shirts that they sold
Let y represent the number of hats that they sold
They sold the T-shirts for $12 and the Hats for $5. This means that the cost of x T shirts and y hats would be
12 * x + 5 * y
= 12x + 5y
The total amount raised was $265. It means that
12x + 5y = 265 equation 1
Also, the total number of t shirts and hats sold was 32. It means that
x + y = 32
x = 32 - y
Substituting x = 32 - y into equation 1, it becomes
12(32 - y) + 5y = 265
384 - 12y + 5y = 265
- 12y + 5y = 265 - 384
7y = 119
y = 119/7
y = 17
x = 32 - y = 32 - 17
x = 15
15 T shirts and 17 hats
A triangle has two sides of length 3 and 3. What value could the length of thethird side be? Check all that apply.A. 6B. 12C. 5D. 8E4F. 2
To find out the length of the third segments of the triangle, we can use this formula.
[tex]\begin{gathered} a^2=b^2+c^2 \\ a^2=3^2+3^2 \\ a^2=9+9 \\ a^2=18 \\ a=4.25 \end{gathered}[/tex]The, third length is very close to the 4. thus, the option (E) is correct.
Can you please help me figure out how to do this?
Given
[tex]f(x)=2x^2-4x-3[/tex]
Procedure
[tex]\begin{gathered} f(-1)=2\cdot(-1)^2-4\cdot(-1)-3 \\ f(-1)=2+4-3 \\ f(-1)=3 \end{gathered}[/tex]
The answer would be f(-1) = 3
If the measure of one complementary angle is 30° more than twice the other angle measure, writean equation and find the measure of each angle.
For this problem we kow that the measure of one complementary angle is 30º more than twice the other angle measure
If our original angle is xthe complement would be 90-xº. then using the statement we can write the following equation:
[tex]x=2(30+90-x)[/tex]And from this equation we can solve for x like this:
[tex]x=240-2x[/tex]Adding 2x in both sides we got:
[tex]3x=240[/tex]And dividing both sides by 3 we got:
[tex]x=\frac{240}{3}=80º[/tex]And the final answer for this case would be 80º
Using a standard 52-card deck, Michelle will draw 6 cards with replacement. If Event A = drawing all hearts and Event B =drawing no face cards, which of the following best describes events A and B?
The described events can be classified as independent.
Mainly because the probability of one event won't change the probability of the other event.
Hence, the answer is independent.
24. Simplify x^2 + 7x + 12 x + 3
We reduce like terms:
[tex]\begin{gathered} x^2+7x+12x+3 \\ x^2+19x+3 \end{gathered}[/tex]therefore, the answer is x^2+19x+3
Write an explicit formula for An, the n' term of the sequence 14, 10, 6,..
An explicit formula for the arithmetic sequence is aₙ = 14 - 4(n - 1).
How to write an explicit formula for the arithmetic sequence?Mathematically, the nth term of an arithmetic sequence can be calculated by using this mathematical expression:
aₙ = a₁ + (n - 1)d
Where:
d represents the common difference.a₁ represents the first term of an arithmetic sequence.n represents the total number of terms.From the information provided, we have the following parameters:
First term, a₁ = 14
Second term, a₂ = 10
Third term, a₃ = 6
Next, we would determine the common difference as follows:
Common difference, d = a₂ - a₁
Common difference, d = 10 - 14
Common difference, d = -4
Substituting the parameters into the mathematical expression, we have;
aₙ = 14 + (n - 1)(-4)
aₙ = 14 - 4(n - 1).
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please help! so confused and every tutor keeps dropping my question
First of all, we see that this curve is indeed a function of x.
A function, by definition, assigns exactly one value (generally called y) for each x in the domain.
For a continuous domain like this, if we pass a vertical line through the graph, and this line touches exactly one point at a time, then this graph represents a function of x. And this happens for the given graph.
For the second part, we need to determine the domain and range of this function.
The domain consists of all the values of x for which the function is defined. When it has a filled ball at an ending point of the graph, this means the domain is closed in that point, that is, the x-coordinate of this ending point belongs in the domain.
In this case, for interval notation, we use square brackets to represent the domain - "[" or "[".
When we have a point with an empty ball, on the other hand, the x-coordinate of that point doesn't belong in the domain, and we use parentheses - "(" or ")".
Now, concerning the graph in this question, we see that both endings have filled balls. So, both -3 and 2 (the x-coordinates of these points) belong in the domain.
Therefore, in interval notation, the domain of this function is:
[-3, 2]
Finally, the range is formed by all values of y that are reached by the graph, from the smallest to the larger (global minimum and maximum of the function).
Therefore, the range of this function is:
[-3, 3]
Notice that we also use square brackets to represent the range, since both points with y-coordinates -3 and 3 belong in the graph.