A cosine function has the form
[tex]y=A\cdot\cos (Bx+C)+D[/tex]Where A is the amplitude, B is 2pi/T, and C is null in this case because the phase is not being specified, and D is the vertical shift (midline).
Using all the given information, we have
[tex]y=3\cdot\cos (\frac{2\pi}{T}x)+4[/tex]Then,
[tex]y=3\cdot\cos (\frac{2\pi}{\frac{8}{5}}x)+4=3\cdot\cos (\frac{10\pi}{8}x)+4=3\cdot\cos (\frac{5\pi}{4}x)+4[/tex]Hence, the function is
[tex]y=3\cos (\frac{5\pi}{4}x)+4[/tex]An actor invests some money at 7%, and $24000 more than three times the amount at 11%. The total annual interest earned from the investment is $27040. How much did he invest at each amount? Use the six-step method.
0.07x+0.11(3x+24000)=27040
we will solve for x
x=61,000 [ investment at 7%]
Investment at 11% = 3x + 24000
= 3(61000)+24000
= 207000 [ investment at 11%]
when doing right triangle trigonometry how do you determine which sine you use like sin, cos etc?
Let's draw a right triangle to guide us:
Every right triangle will have one hypotenuse side and two leg sides. The hypotenuse is always the bigger one and it is always opposite to the right angle, so in this triangle the hypotenuse is a (the letter can change from exercise to exercise, but it is always the opposite to the rignt angle).
The legs can be classified as adjancent or opposite legs, but this is with respect to the angle we are using.
So, if we are using angle C, the opposite leg is the leg that is opposite to angle C, that is, c.
Thus, the adjancent leg is the leg that is touching the angle C, that is, b.
So, with respect to angle C, we have:
Hypotenuse - a
Opposite leg - c
Adjacent leg - b
The sine is the ratio between the opposite leg and the hypotenuse, always.
The cosine is the ratio between the adjacent leg and the hypotenuse, always.
The tangent is the ratio between the opposite leg and the adjacent leg, always.
For, for angle C, we have:
[tex]\begin{gathered} \sin C=\frac{c}{a} \\ \cos C=\frac{b}{a} \\ \tan C=\frac{c}{b} \end{gathered}[/tex]For angle B, we do the same, however now, the legs are switched, because the leg that is opposite to angle B is b and the leg that is adjance to angle B is c, so, for angle B:
Hypotenuse - a
Opposite leg - b
Adjacent leg - c
And we follow the same for sine, cosine and tangent but now for angle B and with the legs switched:
[tex]\begin{gathered} \sin B=\frac{b}{a} \\ \cos B=\frac{c}{a} \\ \tan B=\frac{b}{c} \end{gathered}[/tex]Questions regaring these ratios normally will present 2 values and ask for a third value. One of the values will be an angle, the other will be side (usually). So, we need to identify which angle are we working with and which sides are the hypotenuse, the opposite leg and adjancent leg with respect to the angle we will work with. Then we identify which of the side we will use and pick the ratio thet relates the sides we will use.
Ariana is going to invest $62,000 and leave it in an account for 20 years. Assuming
the interest is compounded continuously, what interest rate, to the nearest tenth of a
percent, would be required in order for Ariana to end up with $233,000?
The rate of interest that Ariana should get in order to end up with a final amount of $233,000 is 0.07%.
What is compound interest and how is it calculated?
The interest that is calculated using both the principal and the interest that has accrued during the previous period is called compound interest. It differs from simple interest in that the principal is not taken into account when determining the interest for the subsequent period with simple interest.
Mathematically, A = P (1 + (R/f))ⁿ ;
where A = amount that the depositor will receive
P = initial amount that the depositor has invested
R = rate of interest offered to the depositor
f = frequency of compounding offered per year
n = number of years.
Given, Amount that Ariana wants to end up receiving = A = $233,00
Principal amount that Ariana can invest = P = $62,000
Frequency of compounding offered per year = f = 1
Number of years = 20
Let the rate of interest offered to the depositor be = R
Following the formula established in the literature, we have:
233000 = 62000(1 + R)²⁰ ⇒ 3.76 = (1 + R)²⁰ ⇒ 1.07 = 1 + R ⇒ R = 0.07%
Thus, the rate of interest that Ariana should get in order to end up with a final amount of $233,000 is 0.07%.
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There is a 50% chance of rain here and a 10% chance of rain on Mars. Therefore, there is a 45% chance that it will rain in neither place.
The given statement is true.
This is a question of probability.
It is given in the question that:-
Chance of raining here = 50 %
Chance of raining on Mars = 10 %
The given statement is :-
There is a 45 % chance that it will rain in neither place.
Chance of not raining here = 100 - 50 % = 50 % = 1/2
Chance of not raining on Mars = 100 - 10% = 90 % = 9/10
Hence, chance of raining in neither place = (1/2)*(9/10) = 9/20
9/20 = (9/20)*100 = 45 %.
Hence, the given statement "There is a 45% chance that it will rain in neither place" is true.
To learn more about probability, here:-
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Write an equation in slope-intercept form of the line that passes through the given point and is parallel to the graph of the given equation. (-6, -6); y=-2x+4
Answer:
y = 2x + 6
Step-by-step explanation:
Parallel lines have the same slope, so the slope is 2.
y = mx + b
When need the slope which is given to be 2
We will use the point given (-6,-6) for an x and y on the line
m= 2
x -= -6
y = -6
y=mx+ b
-6 = 2(-6) + b Sole for b
-6 = -12 + b Add 12 to both sides
6 = b
y = 2x + 6
How far apart, in inches, would the same two cities be on a map that has a scale of 1 inch to 40 miles?
Using scales, the distance of the two cities on the map would be of:
distance on the map = actual distance/40
What is the scale of a map?A scale on the map represents the ratio between the actual length of a segment and the length of drawn segment, hence:
Scale = actual length/drawn length
In this problem, the scale is of 1 inch to 40 miles, meaning that:
Each inch drawn on the map represents 40 miles.
Then the distance of the two cities on the map, in inches, would be given as follows:
distance on the map = actual distance/40.
If the distance was of 200 miles, for example, the distance on the map would be of 5 inches.
The problem is incomplete, hence the answer was given in terms of the actual distance of the two cities. You just have to replace the actual distance into the equation to find the distance on the map.
A similar problem, also involving scales, is given at brainly.com/question/13036238
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How to write slope intercept form
Answer:
See below
Step-by-step explanation:
If you are given slope (m) and intercept (b) , then write the line equation like this:
y = mx + b
the measure of angle is 15.1 what is measure of a supplementary angle
we get that measure of the supplemantary angle is:
[tex]180-15.1=164.9[/tex]Which two ratios are NOT equal? 1:6 and 3:18 OB. 2:14 and 3:42 OC. 12:6 and 2:1 OD 3:11 and 6:22
Let's check the ratios:
[tex]\begin{gathered} \frac{1}{6} \\ \text{and} \\ \frac{3}{18} \\ \end{gathered}[/tex]First one is already reduced. Let's reduce the 2nd fraction by dividing top and bottom by 3, so
[tex]\frac{3}{18}=\frac{1}{6}[/tex]So, they are equal.
Next ratio:
[tex]\begin{gathered} \frac{2}{14}\text{and}\frac{3}{42} \\ \end{gathered}[/tex]Let's divide both top and bottom by 2 (1st fraction) and top and bottom by (3) in 2nd fraction:
[tex]\begin{gathered} \frac{2}{14}=\frac{1}{7} \\ \text{and} \\ \frac{3}{42}=\frac{1}{14} \end{gathered}[/tex]They aren't equal. So, we have already found our answer.
OB. 2:14 and 3:42 --- is our answer.
If the number of college professors is P and the number of students S, and there are 20 times more students as professors, write an algebraic equation that shows the relationship
Answer
Algebraic equation that shows the relationship is
P = 20S
Explanation
Number of college professors = P
Number of students = S
There are 20 times as many students as professors.
P = (S) (20)
P = 20S
Hope this Helps!!!
a box of cereal states that there are 75 calories in a 3/4 serving what is the unit rate for calories cup how many calories are there in 2 cups
We know that a box of cereal states that there are 75 calories in a 3/4 cup.
To find the unit rate for calories cup we must represent the the situation with an equation
[tex]\frac{75\text{ calories}}{\frac{3}{4}\text{ cup}}=\frac{x\text{ calories}}{1\text{ cup}}[/tex]Then, to find the unit rate for calories we need to solve the equation for x
[tex]x\text{ calories}=\frac{75\text{ calories}\cdot1\text{ cup}}{\frac{3}{4}\text{ cup}}=100\text{ calories}[/tex]Now, to find how many calories there are in 2 cups we must multiply the unit rate for calories by 2
[tex]x\text{ calories=100 calories}\cdot2=200\text{ calories}[/tex]Finally, the answers are:
- The unit rate for calories is 100 calories/cup.
- In 2 cups there are 200 calories.
(x^2+9)(x^2-9) degree and number of terms
ANSWER
Degree: 4
Number of terms: 2
EXPLANATION
Subtract. Write fractions in simplest form. 12/7 - (-2/9) =
You have to subtract the fractions:
[tex]\frac{12}{7}-(-\frac{2}{9})[/tex]You have to subtract a negative number, as you can see in the expression, both negatives values are together. This situation is called a "double negative" when you subtract a negative value, both minus signs cancel each other and turn into a plus sign:
[tex]\frac{12}{7}+\frac{2}{9}[/tex]Now to add both fractions you have to find a common denominator for both of them. The fractions have denominators 7 and 9, the least common dneominator between these two numbers is the product of their multiplication:
7*9=63
Using this value you have to convert both fractions so that they have the same denominator 63,
For the first fraction 12/7 multiply both values by 9:
[tex]\frac{12\cdot9}{7\cdot9}=\frac{108}{63}[/tex]For the second fraction 2/9 multiply both values by 7:
[tex]\frac{2\cdot7}{9\cdot7}=\frac{14}{63}[/tex]Now you can add both fractions:
[tex]\frac{108}{63}+\frac{14}{63}=\frac{108+14}{63}=\frac{122}{63}[/tex]I had $70 and my mother gave me $10 and my father gave me $30 and aunt and uncle gave me $150 and I had another $7 how much do I have
Initial money = 70
then add
10 + 30 + 150 + 7 = 197
Now add both results
70 + 197 = 267
Answer is
You have $267
I need help to solve by using the information provided to write the equation of each circle! Thanks
Explanation
For the first question
We are asked to write the equation of the circle given that
[tex]\begin{gathered} center:(13,-13) \\ Radius:4 \end{gathered}[/tex]The equation of a circle is of the form
[tex](x-a)^2+(y-b)^2=r^2[/tex]In our case
[tex]\begin{gathered} a=13 \\ b=-13 \\ r=4 \end{gathered}[/tex]Substituting the values
[tex](x-13)^2+(y+13)^2=4^2[/tex]For the second question
Given that
[tex](18,-13)\text{ and \lparen4,-3\rparen}[/tex]We will have to get the midpoints (center) first
[tex]\frac{18+4}{2},\frac{-13-3}{2}=\frac{22}{2},\frac{-16}{2}=(11,-8)[/tex]Next, we will find the radius
Using the points (4,-3) and (11,-8)
[tex]undefined[/tex]Consider the line segment porque shown. For which of the following transformations would the image porque be contained entirely in Quadrant II?
We will have the following:
In order to have PQ entirely in the quadrant II, the transformation must be:
*Translate PQ up 4 units and to the left 3 units. [Option K]
What is the value of the expression below when z6?9z + 8
Hello!
Let's solve your expression:
[tex]9z+8[/tex]Let's replace where's z by 6, look:
[tex]\begin{gathered} (9\cdot z)+18 \\ (9\cdot6)+18 \\ 54+18 \\ =72 \end{gathered}[/tex]So the value of this expression when z=6 is 72.
which of the following equations represent a line that is perpendicular to y=-3x+6 and passes through the point (3,2)
Answer:
y = [tex]\frac{1}{3}[/tex] x + 1
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = - 3x + 6 ← is in slope- intercept form
with slope m = - 3
given a line with slope m then the slope of a line perpendicular to it is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{-3}[/tex] = [tex]\frac{1}{3}[/tex] , then
y = [tex]\frac{1}{3}[/tex] x + c ← is the partial equation
to find c substitute (3, 2 ) into the partial equation
2 = 1 + c ⇒ c = 2 - 1 = 1
y = [tex]\frac{1}{3}[/tex] x + 1 ← equation of perpendicular line
Mathematics literacy Finance Break-even analysis homework (1.1 and 1.2 only)
We are given a set of data with the employee number and the corresponding weekly wage.
Part 1.1 To determine the wage per hour we need to find the quotient between the weekly wage and the number of hours worked per week.
In the case of employee 1, we have that his weekly wage was 1680, therefore, the weekly payment per hour is:
[tex]p=\frac{1680}{42}=40\text{ per hour}[/tex]The weekly payment is $40 per hour.
Part 1.2 We have that employee number 4 work a total of 6 hours each day of the week. Since there are 7 days per week we have that the total number of hours during the week is:
[tex]h_4=(6day)(7)=42\text{ }hours[/tex]Now, we multiply by the rate of payment per week, therefore, his weekly pay must be:
[tex]p_4=(42hours)(40\text{ per hour\rparen}=1680[/tex]Therefore, the weekly wage of 4 is 1680.
Part 1.3 To determine the number of hours that employee 8 we must have into account that the number of hours per week by the rate of pay per hour is the total weekly wage, therefore:
[tex](40\text{ per hour\rparen}h_8=2000[/tex]Now, we divide both sides by 40:
[tex]h_8=\frac{2000}{40}=50hours[/tex]Therefore, employee 8 worked 50 hours.
Part 1.4 Since the weekly payment is proportional to the number of hours this means that the employee that worked the least number of hours is the one with the least weekly wage.
We have that employee 5 has the smaller wage, therefore, employee 5 worked the least number of hours.
Part 1.5 we are asked to identify the dependent variable between weekly wage and the number of hours worked.
Since the number of hours does not depend on any of the other considered variables this means that this is the independent variable. Therefore, the dependent variables is the weekly wage. The correct answer is A
Part 1.6 The modal value of a set of data is the value that is repeated the most. We have that the weekly wage that is repeated the most is 1600 since it is the wage of employees 2 and 7. Therefore, the modal value is 1600
Part 1.7 The range of a set of data is the difference between the maximum and minimum values. The maximum wage is 2000 and the minimum is 1160, therefore, the range is:
[tex]R=2000-1160=840[/tex]The range is 840
What are the roots of the function represented by the table?
From the table, the root of the function is a point where y = 0.
Therefore,
The root of the function are ( 4, 0 ) and ( -3, 0 )
Final answer
I and III only Option B
If your distance from the foot of the tower is 20 m and the angle of elevation is 40°, find the height of thetower.
We have to use the tangent of angle 40 to find the height of the tower.
[tex]\text{tan(angle) = }\frac{opposite\text{ side}}{\text{adjacent side}}[/tex]The adjacent side is 20m, and the angle is 40 degrees, then
[tex]\tan (40)\text{ = }\frac{height\text{ of the tower}}{20m}[/tex][tex]\text{height = 20m }\cdot\text{ tan(40) = 20m }\cdot0.84\text{ = }16.8m[/tex]Therefore, the height of the tower is 16.8m
Find the equation of the line parallel to the line y=-1, going through point (-5,4)
In this problem, want to find the equation of a line that will be parallel to a given function through a point.
Recall that parallel lines have the same slope.
We are given the line
[tex]y=-1[/tex]and the point
[tex](-5,4)[/tex]Notice that the equations is technically in slope-intercept form, by the value of the slope will be 0:
[tex]y=0x-1[/tex]Therefore, the slope of the line through (-5,4) will also be zero. We can use that information to find the equation.
Using the form
[tex]y=mx+b[/tex]we can substitute the point and the slope to solve for b:
[tex]\begin{gathered} 4=0(-5)+b \\ \\ 4=b \end{gathered}[/tex]So, the equation of our line is:
[tex]y=0x+4\text{ or }\boxed{y=4}[/tex]I have a calculus question about related rates, pic included
ANSWER
40807 cm³/min
EXPLANATION
The tank has the shape of a cone, with a total height of 9 meters and a diameter of 3.5 m - so the radius, which is half the diameter, is 1.75 m. As we can see, the relationship between the height of the cone and the radius is,
[tex]\frac{r}{h}=\frac{1.75m}{9m}=\frac{7}{36}\Rightarrow r=\frac{7}{36}h[/tex]So the volume of water will be given by,
[tex]V(h)=\frac{1}{3}(\pi r^2)h=\frac{1}{3}\cdot\pi\cdot\frac{7^2}{36^2}h^2\cdot h=\frac{49\pi}{3888}h^3[/tex]Where h is the height of the water (not the tank).
If we derive this equation, we will find the rate at which the volume of water is changing with time,
[tex]\frac{dV}{dt}=\frac{49\pi}{3888}\cdot3h^{3-1}=\frac{49\pi}{3888}\cdot3h^2=\frac{49\pi}{1296}h^2[/tex]We want to know what is the change of volume with respect to time, and this is,
[tex]\frac{dV}{dt}=\frac{dV}{dt}\cdot\frac{dh}{dt}[/tex]Because the height also changes with time. We know that this change is 24 cm per minute when the height of the water in the tank is 1 meter (or 100 cm), so we have,
[tex]\frac{dV}{dt}=\frac{49\pi}{1296}h^2\cdot\frac{dh}{dt}=\frac{49\pi}{1296}\cdot100^2cm^2\cdot\frac{24cm}{1min}\approx28507cm^3/min[/tex]This is the rate at which the water is increasing in the tank. However, we know that there is a leak at a rate of 12300 cm³/min, which means that in fact the water is being pumped into the tank at a rate of,
[tex]28507cm^3/min+12300cm^3/min=40807cm^3/min[/tex]Hence, the water is being pumped into the tank at a rate of 40807 cm³/min, rounded to the nearest whole cm³/min.
3 /17% of a quantity is equal to what fraction of the quantity
Given:
The objective is to find the fraction of 3/17% of the quantity.
Consider the quantity as x. The fraction of 3/17% of the quantity can be calculated as,
[tex]\begin{gathered} =\frac{3}{17}\frac{1}{100}x \\ =\frac{3}{1700}x \end{gathered}[/tex]Hence, the required fraction of quantity is 3/1700 of x.
Christian buys a $3500 computer using an installment plan that requires 17% down and a 3.7% interest rate. How much is the down payment?
1) Gathering the data
$3500 computer
17% down
3.7% interest rate.
2) Since we want to know how much is that down payment, we must turn that 17% into decimal form, then multiply it by the computer value:
17%=0.17
3500 x 0.17 = $595
3) So Christian must pay $595 as the down payment
Use this information to answer the following two questions. Mathew finds the deepest part of the pond to be 185 meters. Mathew wants to find the length of a pond. He picks three points and records the measurements, as shown in the diagram. Which measurement describes the depth of the pond? Hide All Z between 13 and 14 meters 36 m 14 m between 14 and 15 meters between 92 and 93 meters Х ag between 93 and 94 meters
it's letter A. Between 13 and 14 meters
Because one side measure 14, and the height (depth) could not be
higher than 14 meters .
The length of the pond can be calculated using the Pythagorean theorem
length^2 = 36^2 + 14^2
length^2 = 1296 + 196
length^2 = 1492
length = 38.6 m
Factor.2n2 + 7n + 5
The first step to factor this expression is to find its roots (the values of 'n' that makes this expression equals zero)
To find the roots, we can use the quadratic formula:
(Using the coefficients a=2, b=7 and c=5)
[tex]\begin{gathered} n_1=\frac{-b-\sqrt{b^2-4ac}}{2a}=\frac{-7-\sqrt{49-40}}{4}=\frac{-7-3}{4}=\frac{-10}{4}=\frac{-5}{2} \\ n_2=\frac{-b+\sqrt{b^2-4ac}}{2a}=\frac{-7+3}{4}=\frac{-4}{4}=-1 \end{gathered}[/tex]So the roots of the expression are -5/2 and -1. Now, we can write the expression in this factored form:
[tex]\begin{gathered} a(n-n_1)(n-n_2) \\ 2(n+\frac{5}{2})(n+1) \\ (2n+5)(n+1) \end{gathered}[/tex]So the factored form is (2n+5)(n+1)
How long can you lease the car before the amount of the lease is more than the cost of the car
ANSWER:
48 months
STEP-BY-STEP EXPLANATION:
According to the statement we can propose the following equation, where the price of the car is more than or equal to the amount of the lease. Just like this:
Let x be the number of months
[tex]16920\ge600+340x[/tex]We solve for x, just like this:
[tex]\begin{gathered} 600+340x-600\le16920-600 \\ \frac{340x}{340}\le\frac{16320}{340} \\ x\le48 \end{gathered}[/tex]Therefore, for 48 months, the car rental will be lower
Consider the triangles ADB and EDC. Explain how they are similar.
Example: Triangles like ABC and EDC are similar by SAS similarity, because angle C is congruent in each triangle, and AC/EC = BC/DC = 2. By the definition of similarity, it follows that AB/DE = BC/EF = AC/DF = 2.
4)Describe the difference between a sampling error and non-sampling error .
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data:
sampling error and non-sampling error
Step 02:
statistics:
Sampling error:
It is the error that arises in a data collection process as a result of taking a sample from a population rather than using the whole population.
Non-sampling error:
It is the error that arises in a data collection process as a result of factors other than taking a sample.
non-response errors, coverage errors, interview errors, and processing errors