The pumping stations will deliver approximately 8.188 gallons of oil during the 4-hour period from t = 0 to t = 4.
Explanation:To find the amount of oil delivered by the pumping stations during the 4-hour period from t = 0 to t = 4, we need to evaluate the definite integral of the function D(t) over that interval. The function D(t) is given by D(t) = 5t / (1 + 3t). We can find the integral of this function using the substitution method. Let u = 1 + 3t, then du = 3dt. Rearranging this equation, we have dt = du / 3.
Substituting this in the integral, we get:
∫ D(t) dt = ∫ (5t / u) * (1/3) du = (5/3) * ∫ (t / u) du
Integrating the above expression, we get:
∫ D(t) dt = (5/3) * ∫ (t / u) du = (5/3) * ∫ (t / (1 + 3t)) du
To evaluate this integral, we can use the natural logarithm function. We know that ∫ (1/u) du = ln|u| + C, where C is the constant of integration. Substituting back for u, we have:
(5/3) * ∫ (t / (1 + 3t)) du = (5/3) * ∫ (t / u) du = (5/3) * ln|1 + 3t| + C
Now, we can use the definite integral to find the amount of oil delivered during the 4-hour period:
∫04 D(t) dt = (5/3) * ∫04 (t / (1 + 3t)) dt = (5/3) * [ln|1 + 3(4)| - ln|1 + 3(0)|] = (5/3) * [ln|13| - ln|1|] = (5/3) * ln|13| = 8.188
Therefore, the pumping stations will deliver approximately 8.188 gallons of oil during the 4-hour period from t = 0 to t = 4.
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Which is the correct cofunction identity for cos theta
A. csc(90° - theta)
B. sec(90° - theta)
C. sin(90° - theta)
D. cos(90° - theta)
E. tan(90° - theta)
Answer:
C. sin(90° - θ)
Step-by-step explanation:
A trig function of an angle equals the cofunction of the angle's complement.
The cofunction of cosine is sine, and the complement of θ is 90° - θ.
∴ cosθ = sin(90° - θ)
Answer: Option 'C' is correct.
Step-by-step explanation:
Since we have given that
[tex]\cos \theta[/tex]
We need to find the correct cofunction identity for it.
Cofunction identities represent the relationship among the trigonometric functions.
The value of trigonometric function of an angle is equal to cofunction of its complement.
As we know that sine is a complement of cosine.
so, it becomes,
[tex]\cos \theta=\sin(90^\circ-\theta)[/tex]
Hence, Option 'C' is correct.
The time required for an automotive center to complete an oil change service on an automobile approximately follows a normal distribution, with a mean of 19 minutes and a standard deviation of 3.5 minutes. (a) The automotive center guarantees customers that the service will take no longer than 20 minutes. If it does take longer, the customer will receive the service for half-price. What percent of customers receive the service for half-price? (b) If the automotive center does not want to give the discount to more than 2% of its customers, how long should it make the guaranteed time limit?
Answer:
a. 61.41%
b. 27 minutes
Step-by-step explanation:
a: Find the z-score for the situation.
µ = 19
x-bar = 20
σ = 3.5
z = (20 - 19)/3.5 = 0.29
The p-value for z = 0.29 is 0.6141, so 61.41% of people will get this discount
b: They want no more than 2% to get the discount, so they want less than 98% getting the discount. The z-score for 98% (0.98 as a decimal) is 2.055
*You need to look at the chart and find where 0.98 would be. It's between a z-score of 2.05 and 2.06.
The z-score is 2.055 = (x - 19)/3.5 We are solving for the time for this one. So solve for x...
7.1925 = x - 19 (multiply both sides by 3.5)
26.1925 = x (add 19 to both sides)
So 26.1925 minutes, or about 26 minutes, 12 seconds, so round up to 27 minutes because they want less than 2%. I chose 27 minutes because no places give odd wait times like 26 minutes 12 seconds.
A) The percent of customers receive the service for half-price is; 61.41%
B) The time to make the guaranteed limit is; 26 minutes 12 seconds
What is the p-value of the distribution?
A) We are given;
Population mean; µ = 19
Sample mean; x' = 20
Standard deviation; σ = 3.5
Thus, z-score is;
z = (20 - 19)/3.5
z = 0.29
From online p-value from s-score calculator, the p-value for z = 0.29 is;
p = 0.6141 = 61.41%
B) We are told that they now want more than 2% to get the discount. This means that they want less than 98% or 0.98 getting the discount.
The z-score for 0.98 is; z = 2.055
Thus, using z-score formula, we have;
2.055 = (x' - 19)/3.5
x' - 19 = 3.5 * 2.055
x' - 19 = 7.1925
x' = 26.1925 = 26 minutes 12 seconds
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Which is the best estimate for the mass of a desktop computer?
Answer:
I would say about 60 to 70 pounds. But I have no more information from you to better answer your question, so that's all I have right now.
Step-by-step explanation:
The division algorithm states that if p(x) and d(x) are polynomial functions with d left parenthesis x right parenthesis not equals 0 comma and the degree of d(x) is less than or equal to the degree of p(x), then there exist unique polynomial functions q(x) and r(x) such that
Final answer:
The division algorithm for polynomials establishes that a polynomial p(x) can be divided by a non-zero polynomial d(x) (with degree less or equal to p(x)), to yield unique quotient q(x) and remainder r(x), with r(x) having a lower degree than d(x).
Explanation:
The division algorithm in the context of polynomials is a fundamental concept in algebra that stipulates for any two polynomials p(x) and d(x), with d(x) ≠ 0 and the degree of d(x) less than or equal to the degree of p(x), there exist unique polynomials q(x) and r(x) such that p(x) = d(x) × q(x) + r(x).
In this scenario, q(x) is referred to as the quotient and r(x) is the remainder. The degree of the remainder r(x) will always be less than the degree of d(x), following the division algorithm.
When applying this algorithm to special sets of polynomials like Legendre polynomials, additional properties can be observed, such as the roots of the polynomials or specific transformations like the Poisson bracket that could arise in mathematical physics. Moreover, the concept extends to rational functions, which are the quotients of polynomials.
Describe a realistic situation that could cause you or someone you know to have to use money from a financial reserve
Answer:
Step-by-step explanation:
Your car breaks down unexpectedly and you must have it repaired immediately. That's an example of a situation where you might use money from a financial reserve.
Solve with proper methods.
There are 50 habits on a farm. They are increasing at the rate of 12% every month, How many months will it be until there are 1000 rabbits? Show your work.
Answer:
17.85 months
Step-by-step explanation:
original number of rabbits=50
% increase per month=12%
number of months=x
form equation for 1000 rabbits
(112/100) ×50× x = 1000
56x=1000
x=1000/56
x=17.857 months
Remi invests ?600 for 5 years in a saving account. By the end of the 5 years he has received a total of ?75 simple interest. Work out the annual rate of simple interest
Answer:
2.5 %
Step-by-step explanation:
The simple interest formula is
I = Prt
Data:
I = $75
P = $600
t = 5 yr
Calculation:
75 = 600 × r × 5
75 = 3000 r
Divide each side by 3000
r = 75/3000 = 0.025 = 2.5 % APR
The annual percentage rate is 2.5 %.
Final answer:
The annual rate of simple interest that Remi received from his savings account is 2.5%. This was calculated using the simple interest formula I = PRT, rearranged to solve for R, the annual interest rate.
Explanation:
To work out the annual rate of simple interest that Remi received from his savings account, we can use the formula for simple interest:
I = PRT
where I is the total interest earned, P is the principal amount invested, R is the annual interest rate, and T is the time the money is invested in years.
From the question, we know that Remi invested £600 (P=600), received £75 in interest (I=75), and the investment was for 5 years (T=5). We need to find the annual rate (R).
Rearranging the formula to get value of R, we get:
R = I / (PT)
Substituting the given values:
R = 75 / (600 × 5) (× is the multiplication symbol)
R = 75 / 3000
R = 0.025
To Convert decimal to a percentage, we multiply by 100:
R = 0.025 × 100
R = 2.5%
So, the annual rate of simple interest that Remi received is 2.5%.
Find the missing values for the exponential function represented by the table below.
x y
-2 7
-1 10.5
0 15.75
1
2
Answer:
x y
-2 7
-1 10.5
0 15.75
1 23.625
2 35.4375
Step-by-step explanation:
The general equation of the exponential function is [tex]y=ab^x[/tex].
We know from our table that when [tex]x=0[/tex], [tex]y=15.75[/tex]. Let's replace those values in our equation:
[tex]y=ab^x[/tex]
[tex]15.75=ab^0[/tex]
Remember that [tex]b^0=1[/tex], so:
[tex]15.75=a(1)[/tex]
[tex]15.75=a[/tex]
[tex]a=15.75[/tex]
We also know from our table that when [tex]x=-1[/tex], [tex]y=10.5[/tex]. Let's replace the values again:
[tex]y=ab^x[/tex]
[tex]10.5=ab^{-1}[/tex]
But we now know that [tex]a=15.75[/tex], so let's replace that value as well:
[tex]10.5=15.75b^{-1}[/tex]
Remember that [tex]b^{-1}=\frac{1}{b}[/tex], so:
[tex]10.5=\frac{15.75}{b}[/tex]
[tex]10.5b=15.75[/tex]
[tex]b=\frac{15.75}{10.5}[/tex]
[tex]b=1.5[/tex]
Now, we can put it all together to complete our exponential function:
[tex]y=ab^x[/tex]
[tex]y=15.75(1.5)^x[/tex]
To find the missing values, we just need to evaluate our function at [tex]x=1[/tex] and [tex]x=2[/tex]:
- For [tex]x=1[/tex]
[tex]y=15.75(1.5)^x[/tex]
[tex]y=15.75(1.5)^1[/tex]
[tex]y=23.625[/tex]
- For [tex]x=2[/tex]
[tex]y=15.75(1.5)^x[/tex]
[tex]y=15.75(1.5)^2[/tex]
[tex]y=35.4375[/tex]
Write these numbers in order from least to greatest. ( 3-3/10), (3.1) , (3-1/4)
Our three numbers are...
3 3/10 = 3.3
3.1
3 1/4 = 3.25
So, if we order those from least to greatest, we have...
3.1, 3.25, 3.3
which, in the forms given, is...
3.1, 3-1/4, 3-3/10
What is the volume of a rectangular prism with the dimensions: base 3 1 2 cm, height 1 1 2 cm, and length 5 1 2 cm?
Answer:
The volume of a rectangular prism is [tex]28\frac{7}{8}\ cm^{3}[/tex]
Step-by-step explanation:
we know that
The volume of the rectangular prism is equal to
[tex]V=BHL[/tex]
Convert the given dimensions to an improper fractions
[tex]B=3\frac{1}{2}\ cm=\frac{3*2+1}{2}=\frac{7}{2}\ cm[/tex]
[tex]H=1\frac{1}{2}\ cm=\frac{1*2+1}{2}=\frac{3}{2}\ cm[/tex]
[tex]L=5\frac{1}{2}\ cm=\frac{5*2+1}{2}=\frac{11}{2}\ cm[/tex]
substitute in the formula
[tex]V=(\frac{7}{2})(\frac{3}{2})(\frac{11}{2})=\frac{231}{8}\ cm^{3}[/tex]
Convert to mixed number
[tex]\frac{231}{8}=\frac{224}{8}+\frac{7}{8}=28\frac{7}{8}\ cm^{3}[/tex]
HELP!! Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).
A professor is examining a new strain of bacteria. The amount of bacteria can be modeled by function s(n) = 20 · bn, where n is the number of hours and b is an unknown positive base.
Based on the model, there were initially (answer space) bacteria.
If b = 1.85, the hourly percent growth rate of the bacteria would be
(answer space) %.
Answer:
20
85%
Step-by-step explanation:
You are given the function [tex]S(n)=20\cdot b^n.[/tex]
If n is the number of hours, then initially n=0 and
[tex]S(0)=20\cdot b^0=20\cdot 1=20.[/tex]
If S(n) is the function of exponential growth, then it can be represented as
[tex]S(n)=I\cdot (1+r)^n,[/tex]
where I is the initial amount, r -is the percent growth rate and n is the number of hours.
If b = 1.85, we can represent it as b = 1 + 0.85. Thus, the hourly percent growth rate of the bacteria would be 0.85=85%.
s(n) = 20b^n
n is the time in hours. At the beginning, the time is zero hours, so n = 0.
s(0) = 20 * b^0
s(0) = 20 * 1
s(0) = 20
The initial amount was 20.
For b = 1.85,
s(n) = 20(1.85)^n
s(n) = 20(1 + 0.85)^n
The hourly growth is 0.85.
0.85 * 100% = 85%
The hourly percent change is 85%.
Please answer this question, will give brainliest!
Answer:
9.9 cm
Step-by-step explanation:
We can use the Pythagorean theorem to find the length of CY
a^2 + b^2 = c^2
CY^2 + YZ^2 = CZ^2
YZ = XY since CZ is the perpendicular bisector
YZ = 5
CZ = 7
CY ^2 + 5^2 = 7^2
CY^2 +25 = 49
Subtract 25 from each side
CY^2 = 49-25
CY^2 = 24
Take the square root of each side
sqrt(CY^2) = sqrt(24)
CY = 4.898979
CY = 4.9 cm
We want the length of WY
WY = WC + CY
WC is a radius which is 5 cm
WY = 5cm + 4.9cm
WY = 9.9 cm
Answer:
We should work backwards we need to find YC+CW to get YW
angle bisector theorem means that ZC and XC are equal
then we can use the Pythagorean theorem to get YC
5^2 + x = 7^2
YC= √13
CW = 7 because they are both the radius of a circle
YW= 7+√13
YW=10.60 (rounded)
How to find the final price of a 120$ pair of shoes discounted by 20% and the by 5%
Please answer this question, will give brainliest!
ANSWER
[tex] \angle \: KGI =50 \degree[/tex]
[tex]\angle \: KJI = 130 \degree[/tex]
EXPLANATION
Angles in the same segment are equal.
This implies that;
[tex] \angle \: KGI = \angle \:H[/tex]
Hence,
[tex] \angle \: KGI =50 \degree[/tex]
Also GJKI is a cyclic quadrilateral.
The opposite angles of a cyclic quadrilateral sums up to 180°.
This means that,
[tex] \angle \: KJI + \angle \: KGI = 180 \degree[/tex]
[tex] \angle \: KJI +50 \degree = 180 \degree[/tex]
[tex]\angle \: KJI = 180 \degree - 50 \degree[/tex]
[tex]\angle \: KJI = 130 \degree[/tex]
Solve the quadratic equation by square roots:
1. 4x²= 81
2. (4x-3)²+7=39
Answer:
see explanation
Step-by-step explanation:
(1)
given
4x² = 81 ( divide both sides by 4 )
x² = [tex]\frac{81}{4}[/tex] ( take the square root of both sides )
x = ± [tex]\sqrt{\frac{81}{4} }[/tex] = ± [tex]\frac{9}{2}[/tex]
---------------------------------------------------------------------
(2)
subtract 7 from both sides
(4x - 3)² = 32 ( take the square root of both sides )
4x - 3 = ± [tex]\sqrt{32}[/tex] = ± 4[tex]\sqrt{2}[/tex]
Add 3 to both sides
4x = 3 ± 4[tex]\sqrt{2}[/tex] ( divide both sides by 4 )
x = [tex]\frac{3}{4}[/tex] ± [tex]\sqrt{2}[/tex]
Identify the graph of 2x^2+2y^=9 for theta=30º and write and equation of the translated or rotated graph in general form.
Answer:
The answer is circle; (x')² + (y')² - 4 = 0
Step-by-step explanation:
* At first lets talk about the general form of the conic equation
- Ax² + Bxy + Cy² + Dx + Ey + F = 0
∵ B² - 4AC < 0 , if a conic exists, it will be either a circle or an ellipse.
∵ B² - 4AC = 0 , if a conic exists, it will be a parabola.
∵ B² - 4AC > 0 , if a conic exists, it will be a hyperbola.
* Now we will study our equation:
* 2x² + 2y² = 8
∵ A = 2 , B = 0 , C = 2
∴ B² - 4AC = (0) - 4(2)(2) = -16 < 0
∵ B² - 4AC < 0
∴ it will be either a circle or an ellipse
* Lets use this note to chose the correct figure
- If A and C are equal and nonzero and have the same sign,
then the graph is a circle.
- If A and C are nonzero, have the same sign, and are not equal
to each other, then the graph is an ellipse.
∵ A = 2 and C = 2
∴ The graph is a circle.
∵ D and E = 0
∴ The center of the circle is the origin (0 , 0)
∵ Ф = 30°
∴ The point (x , y) will be (x' , y')
- Where x = x'cosФ - y' sinФ and y = x'sinФ + y'cosФ
∴ x = x'cos(30°) - y'sin(30°)
∴ y = x'sin(30°) + y'cos(30°)
∴ x = (√3/2)x' - (1/2)y' and y = (1/2)x' + (√3/2)y'
∴ [tex]x=\frac{\sqrt{3}x'-y'}{2}[/tex]
∴ [tex]y=\frac{x'+\sqrt{3}y'}{2}[/tex]
* Lets substitute x and y in the first equation
∴ [tex]2(\frac{\sqrt{3}x'-y'}{2})^{2}+2(\frac{x'+\sqrt{3}y'}{2})^{2}=8[/tex]
* Use the foil method
∴ [tex]2(\frac{3x'^{2}-2\sqrt{3}x'y'+y'^{2}}{4})+2(\frac{x'^{2}+2\sqrt{3}x'y'+3y'^{2}}{4})=8[/tex]
* Open the brackets
∴ [tex]\frac{3x'^{2}-2\sqrt{3}x'y'+y'^{2}+x'^{2}+2\sqrt{3}x'y'+3y'^{2}}{2}=8[/tex]
* Collect the like terms
∴ [tex]\frac{4x'^{2}+4y'^{2}}{2}=8[/tex]
* Simplify the fraction
∴ 2(x')² + 2(y')²= 8
* Divide each side by 2
∴ (x')² + (y')² = 4
∴ The equation of the circle is (x')² + (y')² = 4
* The general equation of the circle is (x')² + (y')² - 4 = 0
after rotation 30° about the origin
* Look to the graph
- The blue circle for the equation 2x² + 2y² = 8
- The blue circle for equation (x')² + (y')² - 4 = 0
* That is because the two circles have same centers and radii
- The green line is x' and the purple line is y'
Answer:
The answer is D
Good luck on the Ed-genuity test
Evaluate the expression under the given conditions. sin(θ − ϕ); tan(θ) = 4/3 , θ in Quadrant III, sin(ϕ) = − 10/10 , ϕ in Quadrant IV
Answer: -0.6
Step-by-step explanation:
First thing to do is to solve for θ and ϕ from the given information
tan(θ) = 4/3,
θ = tan-¹4/3,
θ = 53.1°
Since tan is positive in quadrant III, θ = 53.1°
Also,
sin(ϕ) = − 10/10 ,
ϕ = sin-¹-1
ϕ = 270°
If ϕ is in the fourth quadrant, that gives 360 - ϕ i.e 360 - 270 = 90°
Substituting the values of θ and ϕ into sin(θ − ϕ), we have;
Sin(53.1 - 90)
= sin (-36.9°)
= -0.6
The given expression sin(θ - ϕ), with tan(θ) = 4/3, θ in Quadrant III, sin(ϕ) = -10/10, and ϕ in Quadrant IV, is evaluated by using trigonometric principles and identities. Upon calculations, sin(θ - ϕ) comes out to be -3/5.
Explanation:The question is asking us to evaluate the expression sin(θ − ϕ), given that tan(θ) = 4/3, θ is in Quadrant III, sin(ϕ) = -10/10, and ϕ is in Quadrant IV. In trigonometry, tan θ = sin θ/cos θ. We have tan θ = 4/3 and we know that in Quadrant III, tangent is positive but sine and cosine are negative. So, we can make a right triangle where the opposite side is 4 (basing this on the absolute value of the tan θ) and the adjacent side is 3. The hypotenuse then, by using Pythagoras theorem, comes out to be 5. Then sin θ = -4/5 and cos θ = -3/5.
For sin(ϕ), we are given that it equals -1. In Quadrant IV, sine is negative and cosine is positive, so cos ϕ = √(1 - (-1)^2) = 0.
Finally, utilizing the formula sin (a ± ß) = sin a cos ß ± cos a sin ß, we plug in our values to come to the solution sin(θ - ϕ) = (sin θ cos ϕ) - (cos θ sin ϕ) = ((-4/5)*0) - ((-3/5)*-1) = -3/5
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Simply the expression (Picture provided)
Answer:
b. [tex]\csc(x)[/tex]
Step-by-step explanation:
The given expression is
[tex]\frac{\sec(x)}{\tan(x)}[/tex]
We express in terms of basic trigonometric ratios to obtain;
[tex]\frac{\frac{1}{\cos(x)} }{\frac{\sin(x)}{\cos(x)} }[/tex]
This is the same as
[tex]\frac{1}{\cos(x)}\div \frac{\sin(x)}{\cos(x)}[/tex]
[tex]\frac{1}{\cos(x)}\times \frac{\cos(x)}{\sin(x)}[/tex]
Cancel out the common factors;
[tex]\frac{1}{\sin(x)}=\csc(x)[/tex]
Answer:
[tex]\frac{secx}{tanx}[/tex] = cscx
Step-by-step explanation:
We have given a trigonometric expression.
[tex]\frac{secx}{tanx}[/tex]
We have to simplify the above expression.
Since, we know that
secx is reciprocal of cosx.
secx = 1/cosx
Tanx is the ratio of sinx and cosx.
Tanx = sinx / cosx
Given expression becomes
[tex]\frac{1/cosx}{sinx/cosx}[/tex]
[tex]\frac{1}{cosx}\frac{cosx}{sinx}[/tex]
[tex]\frac{1}{sinx}[/tex]
[tex]\frac{secx}{tanx}[/tex] = cscx which is the answer.
Find the value of x in the polygon.
Answer:
x = 25 mm
Step-by-step explanation:
The perimeter of a rectangle is given by
P = 2(l+w) where l is the length and w is the width
We know the perimeter is 60 mm and the width is 5 mm and the length is x
60 = 2(x +5)
Divide each side by 2
60/2 = 2/2(x+5)
30 = x+5
Subtract 5 from each side
30-5 = x+5-5
25 =x
x = 25 mm
[tex]\huge\bold\red{Answer}[/tex]
☑The diagram which is shown above is a rectangle.
✍ Perimeter= 60mm
✍ breadth = 5mm
➡ Perimeter of rectangle = 2(l+b)
✍ 60 = 2(l +5)
✍ 60/2 = l+5
✍ 30 - 5 = l
✍ 25 = l
☑ L = 25mm
❣..hope it helps you..❣
Let p(x)=909+50e−x .
What is p(5) ?
Answer:
The answer is 9.6
hope this helps!
Suppose you are a casino owner, and your casino runs a very simple game of chance. The dealer flips a coin. The customer wins $9 from the casino if it comes up heads and loses $10 if it comes up tails. Question 1. Assuming no one is cheating and the coin is fair, if a customer plays twice, what is the chance they make money?
Answer:
Step-by-step explanation:
The outcome of the game is
Toss 1 Toss 2 Result
H H 18
H T -1
T H -1
T T -20
His only winning position is H ---- H
That means he has only a 1/4 chance in winning. He shouldn't play at all. It might be a fair coin, but it is not a fair game.
Solve on the interval [0/2pi]
1-cos(theta) = (1/2)
Answer:
Final answer is [tex]x=\frac{\pi}{3}[/tex] and [tex]x=\frac{5\pi}{3}[/tex].
Step-by-step explanation:
Given equation is [tex]1-\cos\left(\theta\right)=\frac{1}{2}[/tex]
Now we need to find the solution of [tex]1-\cos\left(\theta\right)=\frac{1}{2}[/tex] in given interval [tex][0, 2\pi ][/tex].
[tex]1-\cos\left(\theta\right)=\frac{1}{2}[/tex]
[tex]-\cos\left(\theta\right)=\frac{1}{2}-1[/tex]
[tex]-\cos\left(\theta\right)=-\frac{1}{2}[/tex]
[tex]\cos\left(\theta\right)=\frac{1}{2}[/tex]
which gives [tex]x=\frac{\pi}{3}[/tex] and [tex]x=\frac{5\pi}{3}[/tex] in the given interval.
Hence final answer is [tex]x=\frac{\pi}{3}[/tex] and [tex]x=\frac{5\pi}{3}[/tex].
In the interval [0, 2π], the cosine function takes on the value 1/2 at two specific angles: π/3 and 5π/3.
What is the interval
To solve the equation 1 - cos(theta) = 1/2 on the interval [0, 2π], this is done by:
Subtract 1/2 from both sides of the equation to isolate the cosine term:
1 - cos(theta) - 1/2 = 0
-cos(theta) + 1/2 = 0
So multiply both sides by -1 to get rid of the negative sign: cos(theta) - 1/2 = 0
1/2 can be written as 2/4: cos(theta) - 2/4 = 0
So look for common denominator for the fraction on the left side, and is 4:
(4*cos(theta) - 2)/4 = 0
Then Multiply both sides by 4 to remove the fraction: 4*cos(theta) - 2 = 0
So, add 2 to both sides: 4*cos(theta) = 2
Lastly divide both sides by 4: cos(theta) = 1/2
Therefore, the solutions to the equation are: theta = π/3 and theta = 5π/3.
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#SPJ6
Determine whether the given set s is a subspace of the vector space v.
a. v=mn(r), and s is the subset of all upper triangular matrices.
b. v is the vector space of all real-valued functions defined on the interval (−∞,∞), and s is the subset of v consisting of those functions satisfying f(0)=0.
c. v=c2(i), and s is the subset of v consisting of those functions satisfying the differential equation y′′−4y′+3y=0.
d. v=c1(r), and s is the subset of v consisting of those functions satisfying f′(0)≥0.
e. v=r2, and s is the set of all vectors (x1,x2) in v satisfying 5x1+6x2=0. f. v=p5, and s is the subset of p5 consisting of those polynomials satisfying p(1)>p(0). g. v=mn(r), and s is the subset of all nonsingular matrices.
a) Yes, s is a subspace of v because the three conditions for a subspace are satisfied. b) Yes, s is a subspace of v because the three conditions for a subspace are satisfied. c) Yes, s is a subspace of v because the three conditions for a subspace are satisfied.
Explanation:a. v=mn(r) is the vector space of all m x n matrices with entries in the real numbers. In this case, s is a subspace of v if it satisfies the three conditions: (1) the zero vector is in s, (2) s is closed under vector addition, and (3) s is closed under scalar multiplication. Since the zero matrix is an upper triangular matrix, it satisfies the first condition. If two upper triangular matrices are added, the result will also be an upper triangular matrix, satisfying the second condition. And if an upper triangular matrix is multiplied by a scalar, the result is still an upper triangular matrix, satisfying the third condition. Therefore, s is a subspace of v.
b. In this case, v is the vector space of all real-valued functions defined on the interval (-∞, ∞). The subset s consists of functions satisfying f(0) = 0. Similar to part a, we need to check if s satisfies the three conditions to be considered a subspace. The zero function satisfies f(0) = 0, so it satisfies the first condition. If two functions in s are added, the sum will also have f(0) = 0, satisfying the second condition. And if a function in s is multiplied by a scalar, the result will still have f(0) = 0, satisfying the third condition. Therefore, s is a subspace of v.
c. In this case, v is the vector space of all second-order linear homogeneous differential equations. The subset s consists of functions satisfying y'' - 4y' + 3y = 0. To be considered a subspace, s would need to satisfy the three conditions: (1) the zero element is in s, (2) s is closed under addition, and (3) s is closed under scalar multiplication. The zero function satisfies the differential equation, so it satisfies the first condition. If two functions in s are added, the sum will also satisfy the differential equation, satisfying the second condition. And if a function in s is multiplied by a scalar, the result will still satisfy the differential equation, satisfying the third condition. Therefore, s is a subspace of v.
What is the probability that you draw a diamond or a spade from a standard deck of cards? A) 1 16 B) 1 4 C) 1 2 D) 3 4
Answer:
The answer is 1/2 which i think that's what you meant from C.
Step-by-step explanation:
A standard deck of cards has 52 cards. 13 of these cards are spades and 13 are diamonds.
the answer to your question is c :]
Simplify the equation √3^15
Answer:
the answer is D. 3^7√3
Step-by-step explanation:
For this case we have to define properties of powers and roots:
[tex]\sqrt [n] {a ^ m} = a ^ {\frac {m} {n}}[/tex]
We also have to:
[tex]3 ^ {15} = 3 ^ {14} * 3[/tex]
So:
[tex]\sqrt {3 ^ {15}} = \sqrt {3 ^ {14} * 3} = 3 ^ {\frac {14} {2}} * \sqrt {3} = 3 ^ 7 * \sqrt {3}[/tex]
Answer:
[tex]3 ^ 7 * \sqrt {3}[/tex]
Option D
Find the x-intercept of the function: f(x) = 6^x - 2. Round answer to the nearest hundredth, if necessary.
Answer:
x = log base 4 of 2
Step-by-step explanation:
I'm assuming this is truly an exponential function as you state and not a mistype.
6^x = 2
Take the log of each side:
x · log(6) = log(2)
Divide each side by log(6):
x = log(2) / log(6)
x = 0.30103 / 0.77815
x = 0.3869... (rounded)
If a single six sided die is rolled once, what are the odds that you will roll a number less than 3?
Answer:
1/3
Step-by-step explanation:
1/3 since there are two numbers less than three, and there are 6 possible outcomes, so 2/6 = 1/3.
In 2000 the population of a small village was 2,400. With an annual growth rate of approximately 1.68%, compounded continuously, what will the population be in 2020 according to the exponential growth function?
Answer:
[tex]3,358\ people[/tex]
Step-by-step explanation:
The formula to calculate the exponential growth function is equal to
[tex]f(x)=P(e)^{rx}[/tex]
where
f(x) is the population
P is the population in the year 2000
r is the rate in decimal
x is number of years since 2000
e is the mathematical constant number
we have
[tex]x=2020-2000=20\ years\\ P=2,400\\ r=0.0168[/tex]
substitute in the formula above
[tex]f(x)=2,400(e)^{0.0168*20}[/tex]
[tex]f(x)=3,358\ people[/tex]
Which of the following is a disadvantage to purchasing points
A.they lower monthly payments
B.closing costs are increased
C.An immediate tax break is received
D.the interest rate is lowered for the life of the loan
Apex Answers
Answer:
B - closing costs are increased
Step-by-step explanation:
All of the other options are advantages to purchasing points. You want lower monthly payments (a), a tax break (c), and lower interest rate (d)
Answer:
The correct answer is option B.closing costs are increased.
Step-by-step explanation:
Mortgage points or discount points refers to the fees paid directly to the lender during closing in exchange for a reduced interest rate.
In this way the buyer buys down the rate, that helps in reducing his monthly mortgage payments.
So, option A, C and D are the advantages of purchasing discount points.
Easy question!
Are the two hexagonal prisms similar? if so, what is the scale factor of the first figure to the second figure?
Answer:
yes5 : 2 or 2.5 : 1Step-by-step explanation:
The ratios of corresponding linear dimensions are identical:
5/2 = 15/6 = 2.5
so we can conclude the figures are similar. The scale factor is the ratio of corresponding dimensions:
first : second = 5 : 2 . . . or . . . 2.5 : 1
Answer:
Yes.
Scale factor = 2.5.
Step-by-step explanation:
They are similar because corresponding dimensions are in the same ratio:
2/5 = 6/15.
The scale factor = 5/2 = 2.5.