For
[tex]\begin{gathered} \sin 61=\cos \theta \\ \theta=\cos ^{-1}(\sin 61) \\ \end{gathered}[/tex]For
[tex]\begin{gathered} \cos 17=\sin \theta \\ \theta=\sin ^{-1}(\cos 17) \\ \end{gathered}[/tex]hi can someone help me
This type of function is non linear.
Define function.A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). A function, according to a technical definition, is a relationship between a set of inputs and a set of potential outputs, where each input is connected to precisely one output. You can tell if a relation is a function by looking at the inputs (the x-coordinates) and outputs (the y-coordinates). Keep in mind that each input has only one output in a function. A function is an equation with a single solution for y for each value of x. Each input of a particular type receives exactly one output when using a function.
Given,
f(x) = x²
This type of function is non linear.
The end behavior is:
as x ⇒ ∞ , y ⇒ ∞
x ⇒ -∞ , y ⇒ -∞
The function graphed is f(x) = (x -3)²
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The sum of two numbers is 51. One number is 15 more than the other. What is the smaller number. Try solving this by writing a system of equations and substitution.
Let's convert the given relationships into an equation.
Let's name the two number x and y.
The sum of the two numbers is 51: x + y = 51
One number is 15 more than the other: x = y + 15
Using the equations that we generated from the given relationships, let's determine the value of the two numbers by substitution.
Let's substitute x = y + 15 to x + y = 51.
[tex]\text{ x + y = 51 }\rightarrow\text{ (y + 15) + y = 51}[/tex][tex]\text{ y + 15 + y = 51 }\rightarrow\text{ 2y = 51 - 15}[/tex][tex]\text{ 2y = 36 }\rightarrow\text{ y = }\frac{36}{2}[/tex][tex]\text{ y = 18}[/tex]Since we now get the value of y, y = 18, let's determine the value of x.
[tex]\text{ x = y + 15 }\rightarrow\text{ x = 18 + 15}[/tex][tex]\text{ x = 33}[/tex]Therefore, the value of the two numbers is 18 and 33.
5 cm5 cmThe surface area of the above figure isA. 208.1 cm2B. 225.6 cm2C. 314.2 cm2D. none of the above
It is a cylinder.
1.- Calculate the area of the base and the top
Area = 2*pi*r^2
Area = 2*3.14*5^2
Area = 157 cm^2
Total area of the base and top = 2 x 157 = 314 cm^2
2.- Calculate the perimeter of the circle.
Perimeter = 2*pi*r
Perimeter = 2*3.14*5
Perimeter = 31.4 cm
3.- Calculate the lateral area
Lateral area = 5 x 31.4
Lateral area = 157 cm^2
4.- Calculate the total area = 157 + 314
= 471 cm^2
5.- Result
D. None of the above
A 65 ft tree casts a 13 ft shadow. At the same time of day, how long would the shadow of a 20 ft building be? (Draw a diagram to help you set up a proportion)
height of tree = 65 ft
length of shadow = 13 ft
Let draw a diagram to illustrate the question effectively
The proportion can be set up below
[tex]\begin{gathered} \frac{65}{20}=\frac{13}{x} \\ \text{cross multiply} \\ 65x=260 \\ x=\frac{260}{65} \\ x=4\text{ ft} \end{gathered}[/tex]Th shadow will be 4 ft. do you
Rewrite the equation in Ax+By=C form.Use integers for A, B, and C.y-4=-5(x+1)
The given equation is
[tex]y-4=5(x+1)[/tex]To write the equation in standard form, first, we have to use the distributive property.
[tex]y-4=5x+5[/tex]Now, we subtract 5x and 5 on both sides.
[tex]\begin{gathered} y-4-5x-5=5x+5-5x-5 \\ -5x+y-9=0 \end{gathered}[/tex]Now, we add 9 on each side
[tex]\begin{gathered} -5x+y-9+9=0+9 \\ -5x+y=9 \end{gathered}[/tex]Therefore, the standard form of the given equation is[tex]-5x+y=9[/tex]Where A = -5, B = 1, and C = 9.y=6/5x+9 how would I graph it
To graph this linear function, we can find both intercepts of the function. To achieve this, we need to solve the equation when y = 0 (for this function) (this will be the x-intercept), and then we need to solve the resulting equation for this function when x = 0 (this will be the y-intercept). Then, we will have two points for which we can graph the function - we need to remember that a line is defined by two points.
Then, we can proceed as follows:
1. Finding the x-intercept[tex]y=\frac{6}{5}x+9,y=0\Rightarrow0=\frac{6}{5}x+9[/tex]Then, we have:
a. Add -9 to both sides of the equation:
[tex]\frac{6}{5}x=-9[/tex]b. Multiply both sides of the equation by 5/6:
[tex]\frac{5}{6}\frac{6}{5}x=-9\cdot\frac{5}{6}\Rightarrow x=-\frac{45}{6}=-\frac{15}{2}=-7.5[/tex]Therefore, the x-intercept is (-7.5, 0).
2. Finding the y-interceptWe have that x = 0 in this case. Then, we have:
[tex]y=\frac{6}{5}x+9\Rightarrow y=\frac{6}{5}(0)+9\Rightarrow y=9[/tex]Therefore, the y-intercept is (0, 9).
Now, we have the points (-7.5, 0) and (0, 9), and we can draw both points on the coordinate plane. The line will pass through these two points:
Mariana, who rents properties for a living, measures all the offices in a building she is renting. Size (square meters) Number of offices 60 3 70 2 98 5 X is the size of a randomly chosen office. What is the expected value of X? Write your answer as a decimal.
The expected value formula is
[tex]E=\Sigma x\cdot P(x)[/tex][tex]\begin{gathered} E=60\cdot\frac{3}{10}+70\cdot\frac{2}{10}+98\cdot\frac{5}{10} \\ E=18+14+49 \\ E=81 \end{gathered}[/tex]Hence, the expected value is 81.Mr. and Mrs. Hill hope to send their son to college in fourteen years. How much money should they invest now at an interest rate of 9.5% per year, compounded continuously, in order to be able to contribute $8500 to his education?Round your answer to the nearest cent.
continuouslyUsing the formula for a compounded continously
[tex]P=P_0\cdot e^{r\cdot t}[/tex]where P is the amount on the account after t years compounded at an interest rate r when Po is invested in an account.
then,
[tex]\begin{gathered} 8500=P_0\cdot e^{0.095\cdot14} \\ 8500=P_{0^{}}\cdot e^{1.33} \\ P_0=\frac{8500}{e^{1.33}} \\ P_0=2248.056\approx2248.06 \end{gathered}[/tex]A rectangular field is nine times as long as it is wide. If the perimeter of the field is 1100 feet, whatare the dimensions of the field?The width of the field isfeet.The length of the field isfeet.
Given:
The perimeter of the rectangular field is 1100 feet.
According to the question,
l=9w
To find the dimensions:
Substitute l=9w in the perimeter formula,
[tex]\begin{gathered} 2(l+w)=1100 \\ 2(9w+w)=1100 \\ 20w=1100 \\ w=55\text{ f}eet \end{gathered}[/tex]Since the width of the rectangle is 55 feet.
The length of a rectangle is,
[tex]55\times9=495\text{ f}eet[/tex]Hence,
The width of the rectangle is 55 feet.
The length of a rectangle is 495 feet.
Two balls are drawn in succession without replacement from an urn containing 5 red balls and 6 blue balls. Let Z be the random variable representing the number of blue balls. Construct the probability distribution and histogram of the random variable Z.
ANSWER and EXPLANATION
Let R represent the number of red balls.
Let B represent the number of blue balls.
There are four possible outcomes when the balls are picked:
[tex]\lbrace RR,RB,BR,BB\rbrace[/tex]We have that Z is the random variable that represents the number of blue balls.
This implies that the possible values of Z are:
To construct the probability distribution, we have to find the probabilities of each of the outcomes:
[tex]\begin{gathered} P(RR)=\frac{5}{11}*\frac{4}{10}=\frac{2}{11} \\ P(RB)=\frac{5}{11}*\frac{6}{10}=\frac{3}{11} \\ P(BR)=\frac{5}{11}*\frac{6}{10}=\frac{3}{11} \\ P(BB)=\frac{6}{11}*\frac{5}{10}=\frac{3}{11} \end{gathered}[/tex]Hence, the probabilities for the possible outcomes of the random variable are:
[tex]\begin{gathered} P(Z=0)=\frac{2}{11} \\ P(Z=1)=\frac{3}{11}+\frac{3}{11}=\frac{6}{11} \\ P(Z=2)=\frac{3}{11} \end{gathered}[/tex]Therefore, the probability distribution is:
Now, let us plot the histogram:
That is the answer.
Find an equation of the circle having the given center and radius.Center (-3, 3), radius 16
The equation of a circle is given by the next formula:
[tex](x-h)^2+(y-k)^2=r^2[/tex]Where the center is the point (h, k) and r means the radios. Therefore:
[tex]\begin{gathered} (x-(-3))^2+(y-3)^2=(\sqrt[]{6}_{})^2 \\ (x+3)^2+(y-3)^2=6^{} \end{gathered}[/tex]Answer is letter C
What is the volume of this sphere? Use a ~ 3.14 and round your answer to the nearest! hundredth. 5 m cubic meters
We will have the following:
[tex]V=\frac{4}{3}\pi r^3[/tex]Now, we replace the values and solve:
[tex]V=\frac{4}{3}(3.14)(5)^3\Rightarrow V\approx523.33[/tex]So, the volume of the sphere is approximately 523.33 cubic meters.
***Example with an 8 m radius***
If the radius of the sphere were of 8 meters, we would have:
[tex]V=\frac{4}{3}(3.14)(8)^3\Rightarrow V\approx2143.57[/tex]So, the volume of such a sphere would be approximately 2143.57 cubic meters.
Is this a right triangle?Use the Pythagorean Theorem to find out!20 cm12 cm16 cmYesNo
If this is a right triangle, then the Pythagorean theorem has to be valid.
This means that the sum of the squares of the legs has to be equal to the square of the hypotenuse (NOTE: we can identify the potential hypotenuse by finding the side with the most length).
Then, we calculate:
[tex]\begin{gathered} 12^2+16^2=20^2 \\ 144+256=400 \\ 400=400\longrightarrow\text{True} \end{gathered}[/tex]As the Pythagorean theorem is valid for this side's lengths, we know that this triangle is a right triangle.
Answer: Yes.
the top ten medal winning nations in a in a particular year are shown in the table. use the given information and calculate the median number of bronze medals for all nations round to the nearest tenth as needed
We have the following:
We know that to calculate the average we must add the corresponding values of bronze medals of each nation and then divide by the number of nations like this
[tex]\begin{gathered} m=\frac{11+7+9+5+5+3+8+4+6+0}{10} \\ m=\frac{58}{10} \\ m=5.8 \end{gathered}[/tex]the median number of bronze medals for all nations is 5.8
Find the coordinates of the center, vertices, covertices, foci, length of transverse and conjugate axis and the equation of the asymptotes. Then graph the hyperbola.
The given equation is,
[tex]\frac{x^2}{36}-\frac{y^2}{16}=1\text{ ---(1)}[/tex]It can be rewritten as,
[tex]\frac{x^2}{6^2}-\frac{y^2}{4^2}=1\text{ ---(2)}[/tex]The above equation is similar to the standard equation of left-right facing a hyperbola given by,
c. Where would the line y = - 2x + 1 lie? Again, justify your prediction and add the graph of this lineto your graph from part (b).
Given:
b) First the two lines are graphed,
[tex]\begin{gathered} y=2x+3 \\ y=2x-2 \end{gathered}[/tex]Now, yoshi wants to add one more equation,
[tex]y=2x+1[/tex]The graph is represented as,
In the above graph the green line represents the y=2x+1 and it lies between the line y= 2x+3 and y= 2x-2.
c) The graph of the line y = -2x +1
It is observed that the green line y= -2x+1 intersects both the lines y= 2x+3 and y= 2x-2.
questionSuppose $24,000 is deposited into an account paying 7.25% interest, which is compoundedcontinuouslyHow much money will be in the account after ten years if no withdrawals or additional depositsare made?
This is a compound interest question and we have been given:
Principal (P) = $24000
Rate (r) = 7.25%
Years (t) = 10
However, we are told this value is compounded continuously. This means that for every infinitesimal time period, the value keeps being compounded.
The formula for finding the compound interest is:
[tex]\text{Amount}=P(1+\frac{r}{n})^{nt}[/tex]But because the compounding period is continuous and therefore, infinitesimal,
[tex]\begin{gathered} Amount=P(1+\frac{r}{n})^{nt} \\ But, \\ n\to\infty \\ \\ \therefore Amount=\lim _{n\to\infty}P(1+\frac{r}{n})^{nt} \end{gathered}[/tex]This is similar to the general formula for Euler's number (e) which is:
[tex]e=\lim _{n\to\infty}(1+\frac{1}{n})^n[/tex]Thus, we can re-write the Amount formula in terms of e:
[tex]\begin{gathered} \text{Amount}=\lim _{n\to\infty}P(1+\frac{r}{n})^{nt} \\ \text{This can be re-written as:} \\ \\ Amount=\lim _{n\to\infty}P(1+\frac{r}{n})^{\frac{n}{r}\times r\times t}\text{ (move P out of the limit because it is a constant)} \\ \\ \text{Amount}=P\lim _{n\to\infty}((1+\frac{r}{n})^{\frac{n}{r}})^{r\times t} \\ \\ \text{Amount}=P(\lim _{n\to\infty}(1+\frac{r}{n})^{\frac{n}{5}})^{rt} \\ \\ \text{but,} \\ e=(\lim _{n\to\infty}(1+\frac{r}{n})^{\frac{n}{r}} \\ \\ \therefore\text{Amount}=Pe^{rt} \end{gathered}[/tex]Therefore, we can find the amount of money in the account after 10 years:
[tex]\begin{gathered} \text{Amount}=Pe^{rt} \\ P=24000 \\ r=7.25\text{ \%=}\frac{7.25}{100}=0.0725 \\ t=10\text{ years} \\ \\ \therefore\text{Amount}=24000\times e^{10\times0.0725} \\ \\ \text{Amount}=24000\times2.06473 \\ \\ \therefore\text{Amount}=49553.546\approx49553.55 \end{gathered}[/tex]Therefore the amount after compounding continuously for 10 years is:
$49553.55
I need help part two and three of this question:A line passes through the following points: (6,3) and (2,9)1. Write the equation of the lineWhich I got y=-3/2 x+122. Write an equation of a line that is perpendicular to the original form. 3. Write the equation of a line that is parallel to the original form.
Part 2:
To determine an equation that is perpendicular to the line equation y = -3/2x + 12, get the negative reciprocal of the slope of the line equation.
[tex]\begin{gathered} \text{Given slope: }m=-\frac{3}{2} \\ \\ \text{The negative reciprocal is} \\ -\Big(-\frac{3}{2}\Big)^{-1}=\frac{2}{3} \\ \\ \text{We can now assume that any line in the form} \\ y=\frac{2}{3}x+b \\ \text{where }b\text{ is the y-intercept} \\ \text{is perpendicular to the line }y=-\frac{3}{2}x+12 \end{gathered}[/tex]Part 3:
An equation that is parallel to the line y = -3/2x + 12, is a line equation that will have the same slope as the original line.
Given that the slope of the line is m = -3/2, then any line equation in the form
[tex]\begin{gathered} y=-\frac{3}{2}x+b \\ \text{where} \\ b\text{ is the y-intercept} \end{gathered}[/tex]- 9 = 12 what is the value of K?
For this case we have the following expression given:
k/3 -9 = 12
We can add 9 in both sides and we got:
k/3 = 12+9
k/3= 21
And if we multiply in both sides by 3 we got:
k = 21*3 = 63
Find (w∘s)(x) and (s∘w)(x) for w(x)=7x−2 and s(x)=x^2−7x+5
(w∘s)(x)=
The two composite functions have their values to be (w o s)(x) = 7x² - 49x + 33 and (s o w)(x) = (7x - 2)² - 7(7x - 2) + 5
How to determine the composite functions?Composite function 1
The given parameters are
w(x) = 7x - 2
s(x) = x² - 7x + 5
To calculate (w o s)(x), we make use of
(w o s)(x) = w(s(x))
So, we have
(w o s)(x) = 7s(x) - 2
Substitute s(x) = x² - 7x + 5
(w o s)(x) = 7(x² - 7x + 5) - 2
Expand
(w o s)(x) = 7x² - 49x + 35 - 2
Simplify
(w o s)(x) = 7x² - 49x + 33
Composite function 2
Here, we have
w(x) = 7x - 2
s(x) = x² - 7x + 5
To calculate (s o w)(x), we make use of
(s o w)(x) = s(w(x))
So, we have:
(s o w)(x) = w(x)² - 7w(x) + 5
Substitute w(x) = 7x - 2
(s o w)(x) = (7x - 2)² - 7(7x - 2) + 5
So, the composite functions are (w o s)(x) = 7x² - 49x + 33 and (s o w)(x) = (7x - 2)² - 7(7x - 2) + 5
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Find the average rate of change over the interval 0, 1 for the quadratic function graphed.
the average rate of the change is ,
[tex]=\frac{3-5}{1-0}[/tex][tex]=\frac{-2}{1}=-2[/tex]Katherine bought à sandwich for 5 1/2 dollar and a adrink for $2.60.If she paid for her meal with a $ 10 bill how much money did she have left?
To find out how much Katherin have left we need to substrac the amount she spent:
[tex]10-5\frac{1}{2}-2.6=10-5.5-2.6=1.9[/tex]Therefore she has $1.9 left.
To do this same problem in fraction form we need to convert the 2.6 in fraction, to do this we multiply the number by 10 and divided by ten. Then:
[tex]2.6\cdot\frac{10}{10}=\frac{26}{10}=\frac{13}{5}[/tex]then we have:
[tex]\begin{gathered} 10-5\frac{1}{2}-\frac{13}{5}=10-\frac{11}{2}-\frac{13}{5} \\ =\frac{100-55-26}{10} \\ =\frac{19}{10} \end{gathered}[/tex]Therefore, the answer in decimal form is 19/10 dollars.
In the xy-plane, line n passes through point (0,0) and has a slope of 4. If line n also passes through point (3,a), what is the value of a?
Convert the fraction to a decimal. Round the quotient to hundredths when necessary70 over 45
Given:
[tex]\frac{70}{45}[/tex]Required:
We need to convert the given fraction to a decimal.
Explanation:
Divide the number 70 by 45.
[tex]\frac{70}{45}=1.555...[/tex]Round off to the nearest hundredth.
[tex]\frac{70}{45}=1.56[/tex]Final answer:
[tex]\frac{70}{45}=1.56[/tex]Given:
[tex]\frac{70}{45}[/tex]Required:
We need to convert the given fraction to a decimal.
Explanation:
Divide the number 70 by 45.
[tex]\frac{70}{45}=1.555...[/tex]Round off to the nearest hundredth.
[tex]\frac{70}{45}=1.56[/tex]Final answer:
[tex]\frac{70}{45}=1.56[/tex]Solve for u-6u+3(u-3)=12
Answer: u=7
Given:
[tex]-6u+3(u-3)=12[/tex]- Distribute 3(u-3):
[tex]\begin{gathered} -6u+3(u-3)=12 \\ \Rightarrow-6u+3u-9=12 \end{gathered}[/tex]- Combine like terms:
[tex]\begin{gathered} \begin{equation*} -6u+3u-9=12 \end{equation*} \\ \Rightarrow-6u+3u=12+9 \\ \Rightarrow-3u=21 \end{gathered}[/tex]- Divide both sides by -3:
[tex]\begin{gathered} \begin{equation*} -3u=21 \end{equation*} \\ \Rightarrow\frac{-3u}{-3}=\frac{21}{-3} \\ \Rightarrow u=7 \end{gathered}[/tex]Therefore, u=7.
Ms. Bell's mathematics class consists of 6 sophomores, 13 juniors, and 10 seniors.
How many different ways can Ms. Bell create a 3-member committee of sophomores
if each sophomore has an equal chance of being selected?
The number of different ways in which Ms. Bell's can select 3-member committee of sophomores is 20 ways.
What is termed as the combination?Selections are another name for combinations. Combinations are the selection of items from a given collection of items. We need not aim to arrange anything here. Combinations do seem to be selections made by having taken some or all of a set of objects, regardless of how they are arranged. The amount of combinations of n things taken r at a time is denoted by nCr and can be calculated as nCr=n!/r!(nr)!, where 0 r n.0 ≤ r ≤ n.For the given question;
Ms. Bell's mathematics class consists -
6 sophomores, 13 juniors, and 10 seniors.Ms. Bell create a 3-member committee of sophomores with unbiased outcomes.
The section of 3 sophomores can be done as;
⁶C₃ = 6!/3!(6-3)!
⁶C₃ = 6/3!.3!
⁶C₃ = 20 ways.
Thus, the number of different ways in which Ms. Bell's can select 3-member committee of sophomores is 20 ways.
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Select the correct product of (x + 3)(x - 5). CX - 15 X5 + 3x - 5x2 - 15 X - 15 C x + 3x - 5x - 15
Distributive property:
[tex](a+b)(c+d)=ac+ad+bc+bd[/tex]Multiplication of powers with the same base:
[tex]a^m\cdot a^n=a^{m+n}[/tex]For the given expression:
[tex]\begin{gathered} (x^2+3)(x^3-5)=x^2\cdot x^3+x^2\cdot(-5)+3\cdot x^3+3\cdot(-5) \\ \\ =x^{2+3}-5x^2+3x^3-15 \\ =x^5-5x^2+3x^3-15 \\ =x^5+3x^3-5x^2-15 \end{gathered}[/tex]Answer is the second option
The two-way table shows the number of students that do or do not do chores at home and whether they receive an allowance or not. I Allowance No Allowance 13 3 Do Chores Do Not Do Chores 5 a. How many total students do chores? b. What is the relative frequency of students that do chores and get an allowance to the number of students that do chores? Round to the nearest hundredth if necessary. chores nor get an allowance to the total number of What is the relative frequency of students that do not students? Round to the nearest hundredth if necessary, d. Of those that do not do chores what percentage still receive an allowance?
a) do chores 13 + 3 = 16
answer: 16 students
b) this is
[tex]\frac{chores\text{ and allowance}}{\text{chores}}=\frac{13}{16}=0.8125[/tex]answer: 0.81
c) this is
[tex]\frac{\text{no chores and no allowance}}{total}=\frac{4}{25}=0.16[/tex]answer: 0.16
d) this is
[tex]\frac{no\text{ chores and allowance}}{no\text{ chores}}\times100=\frac{5}{9}\times100=\frac{500}{9}=55.55[/tex]answer: 55.55%
Order the following integers from least to greatest.-41, -53, -73, -78 A. -78, -53, -73, -41 B. -78, -73, -41, -53 C. -73, -78, -53, -41 D. -78, -73, -53, -41
The value of negative integers decreases the further we get from the 0 point on the number line.
Therefore, if we arrange the numbers in ascending order ignoring the negative sign, the numbers will be in descending order when the negative sign is included.
By the definition above, we can say that the smallest number of the lot is -78 and the largest one is -41.
The numbers can be ordered from least to greatest as shown below:
[tex]-78,-73,-53,-41[/tex]OPTION D is the correct answer.
New York City mayor Michael made it his mission to reduce smoking in New York City. New York city’s adult smoking rate is 13.2%. In a random sample of 3932 New York City residents, how many of those people smoke? Round to the nearest integer
519 people smoked
Explanation
to figure out this we need to find teh 13.2 % of 3932
so
Step 1
Convert 13.2% to a decimal by removing the percent sign and dividing by 100
then
[tex]13.2\text{ \%}\rightarrow\frac{13.2}{100}\rightarrow0.132[/tex]Step 2
now, multyply the number by the percentage ( in decimal form),so
[tex]\begin{gathered} 13.2\text{ \% of 3932=0.132}\cdot3932=519.04 \\ \text{rounded} \\ 519 \end{gathered}[/tex]therefore, the answer is
519 people smoked
I hope this helps you