(X^3+2x^2) +(5x+10)
X^2(x+2) +5(x+2)
(X^2+5)(x+2)
Answer:
[tex](x+2)\text{ and }(x^2+5)[/tex] are the factors of given polynomial,
Step-by-step explanation:
We are given the following information in the question:
We are given a polynomial:
[tex]x^3 + 2x^2 + 5x + 10[/tex]
We have to factor this polynomial with the help of grouping method.
Grouping can be dine with the help of taking the common factors.
Working:
[tex]\text{Taking } x^2 \text{ common from the first two terms and 5 from the other two terms, we have}\\\Rightarrow x^2(x + 2) +5(x + 2)\\\text{Aain taking the common term}\\\Rightarrow (x+2)(x^2+5)[/tex]
Hence, the two factors of the given polynomial are:
[tex](x+2)\text{ and }(x^2+5)[/tex]
23 points help asap
Which two categories, when added together, equal 65% of the credit score wheel?
add payment history and total debt it gibes you 65%
PLEASE HELP What is the radical form of the expression 423 ? 43−−√ 34−−√ 42−−√3 24−−√3
For this case we must express the number "423" in a radical way.
We have to, we can rewrite it as:
[tex]423 = 9 * 47 = 3 ^ 2 * 47[/tex]
So, we have to:
[tex]423 = \sqrt {3 ^ 2 * 47}[/tex]
By definition of properties of powers and roots we have:
[tex]\sqrt [n] {a ^ m} = a ^ {\frac {m} {n}}[/tex]
So:
[tex]423 = \sqrt {3 ^ 2 * 47} = 3 \sqrt {47}[/tex]
Answer:
[tex]3 \sqrt {47}[/tex]
Answer:
3^√4^2
Step-by-step explanation:
Please help me out with this question, THANKS!
Answer:
line B.)
Step-by-step explanation:
2x+5y=10
5y=-2x+20
(5y/5) (2x+10/5)
y= (-2/5)x +2
Solve the differential equation by variation of parameters, subject to the initial conditions y(0) = 1, y'(0) = 0. 36y'' − y = xex/6
The homogeneous ODE
[tex]36y''-y=0[/tex]
has characteristic equation
[tex]36r^2-1=0[/tex]
with roots at [tex]r=\pm\dfrac16[/tex], and admits two linearly independent solutions,
[tex]y_1=e^{x/6}[/tex]
[tex]y_2=e^{-x/6}[/tex]
as the Wronskian is
[tex]W(y_1,y_2)=\begin{vmatrix}e^{x/6}&e^{-x/6}\\\\\dfrac16e^{x/6}&-\dfrac16e^{-x/6}\end{vmatrix}=-\dfrac13\neq0[/tex]
Variation of parameters has us looking for solutions of the form
[tex]y_p=u_1y_1+u_2y_2[/tex]
such that
[tex]u_1=-\displaystyle\int\frac{y_2xe^{x/6}}{W(y_1,y_2)}\,\mathrm dx[/tex]
[tex]u_2=\displaystyle\int\frac{y_1xe^{x/6}}{W(y_1,y_2)}\,\mathrm dx[/tex]
We have
[tex]u_1=\displaystyle3\int x\,\mathrm dx=\dfrac{3x^2}2[/tex]
[tex]u_2=\displaystyle-3\int xe^{x/3}\,\mathrm dx=-9e^{x/3}(x-3)[/tex]
and we get
[tex]y_p=\dfrac{3x^2e^{x/6}}2-9e^{x/6}(x-3)[/tex]
The general solution is
[tex]y=y_c+y_p[/tex]
[tex]y=C_1e^{x/6}+C_2e^{-x/6}+\dfrac{3x^2e^{x/6}}2-9e^{x/6}(x-3)[/tex]
The initial conditions tell us
[tex]\begin{cases}1=C_1+C_2+27\\\\0=\dfrac{C_1}6-\dfrac{C_2}6-\dfrac92\end{cases}\implies C_1=\dfrac12,C_2=-\dfrac{53}2[/tex]
so that the particular solution is
[tex]y=\dfrac12e^{x/6}-\dfrac{53}2e^{-x/6}+\dfrac32x^2e^{x/6}-9e^{x/6}(x-3)[/tex]
Find the slope of the line
40 Points!
Answer:
-1/2
Step-by-step explanation:
Rise in X / Rise in Y
1 / 2 = 1/2
The answer to ur question is slope =-1/2
Petra wants to buy a skateboard. The skateboard deck usually costs $37.50, but it is on sale for 20% off. If the sales tax rate is 5.2%, how much will Petra pay for the skateboard deck in all?
A. $30.00
B. $31.56
C. $31.95
D. $39.45
Answer:31.56
Step-by-step explanation:
take 37.50 and divide it by 100
then multiply that by 80
you should get 30,
30 is 80% of 37.50 which is 20% off of the original price
then you divide 30 by 100 and multiply it by 5.2
this will get you 1.56
add that to 30 and you get 31.56
Graph the function. Describe its position relative to the graph of the indicated basic function.
f(x)= 4^x-3 ; relative to f(x)= 4^x
Answer:
The answer is (b) moved right 3 units and moved down 3 units
Step-by-step explanation:
∵ f(x) = 4^x ⇒ red in the graph
∵ f(x) = 4^(x-3) - 3 ⇒ blue in the graph
∵ x becomes x - 3 ⇒ means:
The graph moved right 3 units
∵ f(x) = 4^x becomes 4^(x-3) - 3 ⇒ means:
The graph moved down 3 units
∴ The answer is (b)
The graph of f(x) = 4ˣ - 3 is shifted down 3 units compared to the graph of f(x) = 4ˣ
What is an exponential function?An exponential function is in the form:
y = abˣ
Where y,x are variables, a is the initial value of y and b is the multiplication factor.
The graph of f(x) = 4ˣ - 3 and f(x) = 4ˣ is attached.
The graph of f(x) = 4ˣ - 3 is shifted down 3 units compared to the graph of f(x) = 4ˣ
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To fill a planting bed, Mr.Carver uses 5 buckets of soil to 4 buckets peat moss. He needs to use 324 buckets in all to fill the bed. How many buckets of soil and peat moss will he use?
please help bc i am kristinas little sister :(
Buckets of Soil : Buckets of Peat Moss = 5 : 4
Since Mr Carver needs to use 324 buckets in all,
He needs to use 324 * 5/9 = 180 buckets of Soil and 324 * 4/9 = 144 buckets of Peat Moss.
Mr. Carver will use 180 buckets of soil and 144 buckets of peat moss to fill the planting bed, based on the given ratio of 5 buckets of soil to 4 buckets of peat moss.
The problem at hand revolves around ratios and proportions where Mr. Carver is using a mix of soil and peat moss in a specific ratio to fill a planting bed. Given the ratio of 5 buckets of soil to 4 buckets of peat moss, we need to find the total number of buckets for each that will sum up to 324 buckets. First, we'll find the total number of parts in the ratio by adding 5 (for soil) and 4 (for peat moss), which gives us 9 parts. Since we have the total amount of 324 buckets, we can find the value of one part by dividing 324 by 9, which gives us 36.
Once we have the value of one part, we multiply it by the number of parts for soil and peat moss to find their respective quantities:
Soil: 5 parts x 36 buckets per part = 180 bucketsPeat moss: 4 parts x 36 buckets per part = 144 bucketsTherefore, Mr. Carver will use 180 buckets of soil and 144 buckets of peat moss to fill the planting bed.
Your company has a plan that matches your retirement contributions up to 2% of your salary. Your annual salary is $22,000. You are paid bi-weekly (26 times per year). How much should you contribute to the retirement account each pay period to take full advantage of the company match
Answer:
The contribution is $ [tex]16.92\\[/tex] per biweekly payment
Step-by-step explanation:
Salary received in each of the [tex]26 \\[/tex] pay is
[tex]\frac{2}{100} * \frac{22000}{26} \\= 16.92\\[/tex]
Final answer:
To take full advantage of your company's retirement match, contribute at least $16.92 from each bi-weekly paycheck to your retirement account, which is 2% of your annual salary divided by 26 pay periods.
Explanation:
To determine how much you should contribute to the retirement account each pay period to take full advantage of the company match, you need to calculate 2% of your annual salary and then divide that number by the number of pay periods in a year. Here's the step-by-step calculation:
Calculate 2% of your annual salary ($22,000): 0.02 × $22,000 = $440.Divide the annual match by the number of pay periods: $440 ÷ 26 = approximately $16.92.To fully utilize the company's matching program, you should contribute at least $16.92 from each bi-weekly paycheck to your retirement account.
The graph represents the ways Janelle can win the beanbag toss game. Which describes a way Janelle can win the game? land on round 7 times; land on square 1 time land on round 6 times; land on square 2 times land on round 5 times; land on square 3 times land on round 4 times; land on square 4 times
Answer:
Land on round 4 times; land on square 4 times
Step-by-step explanation:
Answer:
The correct option is 4.
Step-by-step explanation:
In the graph orange region represents the ways Janelle can win the beanbag toss game.
From the given graph it is clear that the related line passes through the points (2,5) and (10,0).
If a line passes through two points, then the equation of line is
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
The equation of related line is
[tex]y-5=\frac{0-5}{10-2}(x-2)[/tex]
[tex]y-5=-\frac{5}{8}(x-2)[/tex]
Add 5 on both the sides.
[tex]y-5+5=-\frac{5}{8}(x)+\frac{5}{4}+5[/tex]
[tex]y=-\frac{5}{8}(x)+\frac{25}{4}[/tex]
The shaded region is above the line, so the sign of inequality is ≥.
The required inequality is
[tex]y\geq -\frac{5}{8}(x)+\frac{25}{4}[/tex]
Check the each point whether it satisfy the inequality of not.
In option (1), the given point is (7,1).
[tex]1\geq -\frac{5}{8}(7)+\frac{25}{4}[/tex]
[tex]1\geq 1.875[/tex]
This statement is false.
In option (2), the given point is (6,2).
[tex]2\geq -\frac{5}{8}(6)+\frac{25}{4}[/tex]
[tex]2\geq 2.5[/tex]
This statement is false.
In option (3), the given point is (5,3).
[tex]3\geq -\frac{5}{8}(5)+\frac{25}{4}[/tex]
[tex]3\geq 3.125[/tex]
This statement is false.
In option (4), the given point is (4,4).
[tex]4\geq -\frac{5}{8}(4)+\frac{25}{4}[/tex]
[tex]3\geq 3.75[/tex]
This statement is true. It means the point (4,4) describes a way Janelle can win the game. Therefore the correct option is 4.
WILL GIVE BRAINLIEST! PLEASE HELP!
What is measure of angle R?
Enter your answer as a decimal in the box. Round only your final answer to the nearest hundredth.
Answer:
[tex] R = 28.07^\circ [/tex]
Step-by-step explanation:
Since you have the lengths of all the sides, you can use sine, cosine, or tangent to find the answer. Let's use the tangent ratio.
For angle R, PQ is the opposite leg, and QR is the adjacent leg.
[tex] \tan R = \dfrac{opp}{adj} [/tex]
[tex] \tan R = \dfrac{8}{15} [/tex]
[tex] R = \tan^{-1} \dfrac{8}{15} [/tex]
[tex] R = 28.07^\circ [/tex]
Answer:
were are the answer selection
Step-by-step explanation:
A small box of raspberries costs $2. A large box of raspberries costs $6. Use this information to complete the sentence below.
What sentence below
Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. Let x denote the length of the side of the square being cut out. Let y denote the length of the base.
(a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes. (Do this on paper. Your teacher may ask you to turn in this work.)
(b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (Do this on paper. Your teacher may ask you to turn in this work.)
(c) Write an expression for the volume V in terms of x and y.
V =
(d) Use the given information to write an equation that relates the variables. (Do this on paper. Your teacher may ask you to turn in this work.)
(e) Use part (d) to write the volume as a function of x.
V(x) =
(f) Finish solving the problem by finding the largest volume that such a box can have.
V = ft3
The largest volume that the box can have is 2.25 ft^3, and it's obtained when x = 0.5 ft. This came from a mathematical analysis of the volume function V(x) = 4x^3 - 12x^2 + 9x.
Explanation:
Given that a square is cut from each corner of the cardboard to form an open box, we can come up with the following: The width of the base, represented by y, would be equal to the initial width of the cardboard (3 ft) minus two times the sides of the squares being cut out (2x), so y = 3 - 2x. Part c: will be the Volume of the box which is given by V = x * y^2, substituting for y from the relation obtained above we get. Part d: V = x * (3 - 2x)^2. Part e: is just the simplification of the equation obtained in part d: V(x) = 4x^3 - 12x^2 + 9x. Part f: To find the maximum volume, we need to find the critical points of the V(x) function, these are obtained by setting the derivative of the function equal to zero, solving for x will give x = 0.5 and x =1.5 but since x > 1.5 would give a negative y, the maximum volume is obtained at x = 0.5, substituting this x into the V(x) equation gives V = 2.25 ft^3.
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Solve the system using elimination. Write the solution as an ordered pair. (1 point) SHOW YOUR WORK FOR FULL CREDIT! (2 points)
5x + 4y = 1
-3x + 4y = -7
By using elimination to subtract the second equation from the first, we solve for x and then substitute x back into one of the original equations to solve for y, finding the solution as the ordered pair (1, -1).
Explanation:To solve the system using elimination, we look at the given equations:
1) 5x + 4y = 1
2) -3x + 4y = -7
The goal is to eliminate one variable to solve for the other. In this case, we can subtract the second equation from the first one since they both have the same coefficient for y, which will eliminate the y variable.
5x + 4y = 1Now that we have found x, we can substitute x into one of the original equations to find y:
5(1) + 4y = 1
5 + 4y = 14y = 1 - 54y = -4y = -1The solution to the system is (1, -1), which is an ordered pair representing the x and y values that satisfy both equations.
Andrew plays 1/4 of a song in 1/8 of a minute. How much time, in minutes, does it take him to play an entire song?
1/4 = 1/8
times by 4 to find the whole song as 1/4 × 4 is 1
1 = 4/8
4/8 = 1/2 of a minute
1. C = 2(pi)r and C = d(pi) are the formulas for finding the _____________ of a circle.
Answer:
Circumference
Step-by-step explanation:
The circumference is the distance around the circle. It relates the number of times the diameter will encircle the circumference as 3.14 or π. As a result, the formulas for the circumference of a circle are C = 2πr and C = πd.
The discriminant of the function is_
Given F= {(0,1), (2,4), (4,6), (6,8)} and G= {(2,5), (4,7), (5,8), (6,9), (7,5)}(F-G) (6)
Answer:
(F-G)(x) = -1
Step-by-step explanation:
The notation (F - G)(x) means to subtract each function when x = 6. According to the sets when x = 6 then F is (6,8) and G is (6,9). To subtract the functions, subtract their output values. (F-G)(x) = 8 - 9 = -1.
What is the value of X to the nearest tenth in the triangle below? the triangle is not drawn to scale
Answer:
x = 13.9
Step-by-step explanation:
With respect to the angle shown, we have the hypotenuse (the side opposite the 90 degree angle) and we have the adjacent side (which is the side labeled x). thus we can use the ratio "cosine" to solve this.
The ratio of cosine is defined as:
[tex]Cos\theta=\frac{Adjacent}{Hypotenuse}[/tex]
Where adjacent and hypotenuse are the respective sides and [tex]\theta[/tex] is the angle
Thus, we can now write:
[tex]Cos\theta=\frac{Adjacent}{Hypotenuse}\\Cos(35)=\frac{x}{17}\\Cos(35)*17=x\\x=13.9[/tex]
The first answer choice is right, x = 13.9
The value of X to the nearest tenth in the triangle is 13.9.
We have the neighboring side, which is the side with the letter x, and the hypotenuse, which is the side across from the 90-degree angle, with regard to the angle displayed.
Therefore, the ratio "cosine" can be used to solve this.
The definition of the cosine ratio is: [tex]Cos \theta[/tex] = Adjacent side / hypotenuse.
where the angle and the corresponding sides are called the hypotenuse and adjacent.
So that we may write now:
[tex]Cos(35)= \frac{x}{17}[/tex]
[tex]Cos(35) \times 17 = x[/tex]
Therefore 13.9 = x .
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What is the slope-intercept form of the function that contains the points (6, 2) and (4, 8)?
Answer:
y = -3x + 20
Step-by-step explanation:
See attached photo for explanation
Two less than 3 times a number is the same as the number plus 10. What is the number?
This should help
3 * a - 2 = a + 10
Find the circumference of the circle in terms of pi? [30]
Answer:
60π
Step-by-step explanation:
If the circle has radius of 30 units, substitute r=30 into the formula C = 2πr.
C = 2π(30)
C = 60π
To find the circumference of the circle shown here, let's start with writing the formula down.
Circumference = 2[tex]\pi[/tex]r
We have to divide 60 by 2 because the radius is always half the diameter. Now, we can plug in 30 for r in the formula.
Now we have
circumference = 2[tex]\pi[/tex]30.
This equals 60[tex]\pi[/tex].
Maria did the work to see if 8 is a solution to the equation s-3=11.
s-3=11
8-3=11
5=11
Does her work show that 8 a solution to the equation?
A.)Yes, because the last line of the work is true.
B.)Yes, because 11 minus 8 equals 3.
C.)No, because 8 is even and the other numbers in the equation are odd.
D.)No, because the last line of the work is not true. 5 and 11 are not equal.
Answer: Option D.
Step-by-step explanation:
The expressions of boths sides of the equation must have the same value Or, in other words, if we call the expresion on the left side of the equation A, and the expression on the right side of the equation B, then:
[tex]A=B[/tex]
Keeping the above on mind, you can see that 5 and 11 are not equal, then:
5≠11
The correct value of s is:
[tex]s-3=11\\s=11+3\\s=14[/tex]
Substituting:
[tex]14-3=11\\11=11[/tex]
Answer:
No, because the last line of the work is not true. 5 and 11 are not equal.
Step-by-step explanation:
Identify the graph of x^2-5x+y^2=3 for theta π/3 and write and equation of the translated or rotated graph in general form.
Answer:
The answer is circle; 2(x')² + 2(y')² - 5x' - (5√3)y' - 6 = 0 ⇒ answer (b)
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:
* x² - 5x + y² = 3
∵ A = 1 , B = 0 , C = 1
∴ B² - 4AC = (0) - 4(1)(1) = -4 < 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 = 1 an d C = 1
∴ The graph is a circle.
* To find its center of the circle lets use
∵ h = -D/2A and k = -E/2A
∵ A = 1 and D = -5 , E = 0
∴ h = -(-5)/2(1) = 2.5 and k = 0
∴ The center of the circle is (2.5 , 0)
* Now lets talk about the equation of the circle and angle Ф
∵ Ф = π/3
- That means the graph of the circle will transformed by angle = π/3
- The point (x , y) will be (x' , y'), where
* x = x'cos(π/3) - y'sin(π/3) , y = x'sin(π/3) + y'cos(π/3)
∵ cos(π/3) = 1/2 and sin(π/3) = √3/2
∴ [tex]y=\frac{\sqrt{3}}{2}x'+\frac{1}{2}y'=(\frac{\sqrt{3}x'+y'}{2})[/tex]
∴ [tex]x=\frac{1}{2}x'-\frac{\sqrt{3}}{2}y'=(\frac{x'-\sqrt{3}y'}{2})[/tex]
* Lets substitute x and y in the equation x² - 5x + y² = 3
∵ [tex](\frac{x'-\sqrt{3}y'}{2})^{2}-5(\frac{x'-\sqrt{3}y'}{2})+(\frac{\sqrt{3}x'-y'}{2})^{2}=3[/tex]
* Lets use the foil method
∴ [tex]\frac{(x'^{2} -2\sqrt{3}x'y'+3y')}{4}-\frac{(5x'-5\sqrt{3}y')}{2}+\frac{(\sqrt{3}x'+2\sqrt{3}x'y'+y'^{2})}{4}[/tex]=3
* Make L.C.M
∴ [tex]\frac{(x'^{2}-2\sqrt{3}x'y'+3y'^{2})}{4}-\frac{(10x'-10\sqrt{3}y')}{4}+\frac{(3x'^{2}+2\sqrt{3}x'y'+y'^{2})}{4} =3[/tex]
* Open the brackets ∴[tex]\frac{x'^{2}-2\sqrt{3}x'y'+3y'^{2}-10x'+10\sqrt{3}y'+3x'^{2}+2\sqrt{3}x'y'+y'^{2}}{4}=3[/tex]
* Collect the like terms
∴ [tex]\frac{4x'^{2}+4y'^{2}-10x'+10\sqrt{3}y'}{4}=3[/tex]
* Multiply both sides by 4
∴ 4(x')² + 4(y')² - 10x' + (10√3)y' = 12
* Divide both sides by 2
∴ 2(x')² + 2(y')² - 5x' + (5√3)y' = 6
∵ h = -D/2A and k = E/2A
∵ A = 2 and D = -5 , E = 5√3
∴ h = -(-5)/2(2) = 5/4 =1.25
∵ k = (5√3)/2(2) = (5√3)/4 = 1.25√3
∴ The center of the circle is (1.25 , 1.25√3)
∵ The center of the first circle is (2.5 , 0)
∵ The center of the second circle is (1.25 , 1.25√3)
∴ The circle translated Left and up
* 2(x')² + 2(y')² - 5x' - (5√3)y' - 6 = 0
∴ The answer is circle; 2(x')² + 2(y')² - 5x' - (5√3)y' - 6 = 0
* Look to the graph
- the purple circle for the equation x² - 5x + y² = 3
- the black circle for the equation (x')² + (y')² - 5x' - 5√3y' - 6 = 0
Answer:
B
Step-by-step explanation:
edge
Find the exact value by using a half-angle identity.
tangent of seven pi divided by eight
tan (7pi/8)
please show step by step
Answer:
The exact form of [tex]\tan(\frac{7\pi}{8})[/tex] is [tex]-\sqrt{2}+1[/tex]
Step-by-step explanation:
We need to find the exact value of [tex]\tan(\frac{7\pi}{8})[/tex] using half angle identity.
Since, [tex]\frac{7\pi}{8}[/tex] is not an angle where the values of the six trigonometric functions are known, try using half-angle identities.
[tex]\frac{7\pi}{8}[/tex] is not an exact angle.
First, rewrite the angle as the product of [tex]\frac{1}{2}[/tex] and an angle where the values of the six trigonometric functions are known. In this case,
[tex]\frac{7\pi}{8}[/tex] can be written as ;
[tex](\frac{1}{2})\frac{7\pi}{4}[/tex]
Use the half-angle identity for tangent to simplify the expression. The formula states that [tex] \tan \frac{\theta}{2}=\frac{\sin \theta}{1+ \cos \theta}[/tex]
[tex]=\frac{\sin(\frac{7\pi}{4})}{1+ \cos (\frac{7\pi}{4})}[/tex]
Simplify the numerator.
[tex]=\frac{\frac{-\sqrt{2}}{2}}{1+ \cos (\frac{7\pi}{4})}[/tex]
Simplify the denominator.
[tex]=\frac{\frac{-\sqrt{2}}{2}}{\frac{2+\sqrt{2}}{2}}[/tex]
Multiply the numerator by the reciprocal of the denominator.
[tex]\frac{-\sqrt{2}}{2}\times \frac{2}{2+\sqrt{2}}[/tex]
cancel the common factor of 2.
[tex]\frac{-\sqrt{2}}{1}\times \frac{1}{2+\sqrt{2}}[/tex]
Simplify,
[tex]\frac{-\sqrt{2}(2-\sqrt{2})}{2}[/tex]
[tex]\frac{-(2\sqrt{2}-\sqrt{2}\sqrt{2})}{2}[/tex]
[tex]\frac{-(2\sqrt{2}-2)}{2}[/tex]
simplify terms,
[tex]-\sqrt{2}+1[/tex]
Therefore, the exact form of [tex]\tan(\frac{7\pi}{8})[/tex] is [tex]-\sqrt{2}+1[/tex]
The exact value of [tex]\( \tan\left(\frac{7\pi}{8}\right) \)[/tex] is [tex]\( \frac{2 - \sqrt{2}}{\sqrt{2}} \)[/tex].
To find the exact value of [tex]\( \tan\left(\frac{7\pi}{8}\right) \)[/tex] using the half-angle identity, we proceed as follows:
1. Identify the appropriate half-angle identity:
The tangent half-angle identity is given by:
[tex]\[ \tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{\sin(\theta)} \][/tex]
2. Apply the identity for [tex]\( \theta = \frac{7\pi}{4} \)[/tex] :
First, determine [tex]\( \theta = \frac{7\pi}{4} \)[/tex] , then find [tex]\( \frac{\theta}{2} \)[/tex].
3. Calculate [tex]\( \cos\left(\frac{7\pi}{4}\right) \)[/tex] and [tex]\( \sin\left(\frac{7\pi}{4}\right) \)[/tex]:
[tex]\[ \cos\left(\frac{7\pi}{4}\right) = \cos\left(\frac{2\pi + \frac{\pi}{4}}{2}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \][/tex]
[tex]\[ \sin\left(\frac{7\pi}{4}\right) = \sin\left(\frac{2\pi + \frac{\pi}{4}}{2}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \][/tex]
4. Apply the half-angle identity:
[tex]\[ \tan\left(\frac{7\pi}{8}\right) = \frac{1 - \cos\left(\frac{7\pi}{4}\right)}{\sin\left(\frac{7\pi}{4}\right)} = \frac{1 - \frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} \][/tex]
5. Simplify:
[tex]\[ \tan\left(\frac{7\pi}{8}\right) = \frac{2 - \sqrt{2}}{\sqrt{2}} \][/tex]
Therefore, [tex]\( \tan\left(\frac{7\pi}{8}\right) = \frac{2 - \sqrt{2}}{\sqrt{2}} \)[/tex].
Using the half-angle identity for tangent, we substituted [tex]\( \theta = \frac{7\pi}{4} \)[/tex] and calculated [tex]\( \cos\left(\frac{7\pi}{4}\right) \)[/tex] and [tex]\( \sin\left(\frac{7\pi}{4}\right) \)[/tex] to find [tex]\( \tan\left(\frac{7\pi}{8}\right) \)[/tex] in exact form.
2x + y = 3
x - 2y = -1
If equation one is multiplied by 2 and then the equations are added, the result is _____.
3x = 2
3x = 5
5x = 5
multiply 2x+y = 3 by 2
4x + 2y = 6
x - 2y = -1
add terms (be careful about signs!)
5x + 0 = 2
5x = 5Answer:
5x=5
Step-by-step explanation:
At t=0, a rock is dropped from rest from the top of a building 256 ft high. With what velocity will it strike the ground? What is it acceleration
Answer:
128 ft / second.
Step-by-step explanation:
It's acceleration is due to gravity and is 32 ft s^-2.
To find the velocity when it hits the ground we use the equation of motion
v^2 = u^2 + 2gs where u = initial velocity, g = acceleration , s = distance.
v^2 = 0^2 + 2 * 32 * 256
v^2 = 16384
v = 128 ft s^-1.
The rock will strike the ground with a velocity of 128 ft/s and will experience an acceleration of 32 ft/s^2.
Explanation:First, we need to calculate the time it takes for the rock to fall to the ground. We can use the equation h = (1/2)gt^2, where h is the height of the building (256 ft) and g is the acceleration due to gravity (32 ft/s^2). Solving for t, we find t = sqrt(2h/g) = sqrt(2(256)/32) = 4 seconds.
Next, we can calculate the velocity of the rock just before it hits the ground. We can use the equation v = gt, where g is the acceleration due to gravity and t is the time it takes to fall (4 seconds). Plugging in the values, we find v = (32 ft/s^2)(4s) = 128 ft/s.
Lastly, the acceleration of the rock is equal to the acceleration due to gravity, which is 32 ft/s^2.
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Find the polar equation of the conic with the focus at the pole, directrix x = 4, and eccentricity 1.
(picture provided)
Answer:
Choice D is correct
Step-by-step explanation:
The eccentricity of the conic section is 1, implying we are looking at a parabola. Parabolas are the only conic sections with an eccentricity of 1.
Next, the directrix of this parabola is located at x = 4. This implies that the parabola opens towards the left and thus the denominator of its polar equation contains a positive cosine function.
Finally, the value of k in the numerator is simply the product of the eccentricity and the absolute value of the directrix;
k = 1*4 = 4
This polar equation is given by alternative D
PLEASE HELP!!!
Is investing $4,000 at an interest rate of 5% (compounded annually) and $4,000 at an interest rate of 7% (compounded annually) always, sometimes, or never the same as investing $8,000 (the total of the two principals) at an interest rate of 6% (compounded annually)? Why or why not? Does it matter how long you leave it in the account?
Explain using words and examples, and justify your answer.
No, it is not the same because in option 1, the interest is calculated based on two different principal amounts and then added together. The principal amounts will be different every year because of the varying interest rates. Since one of the interest rates in option 1 is 7%, the principal will grow at a faster rate because the interest rate is applied to a greater and greater principal amount over time. In option 2, the interest is being calculated only on one principal
Answer:
Step-by-step explanation:
No, they are not always the same - the only time that they will be the same is after the first year:
$4000*1.05 + $4000*1.07 = $8480 = $8000*1.06
From there on, they will diverge. For example after the second year:
$4000*1.05^2 + $4000*1.07^2 = $8989.6
$8000*1.06^2 = $8988.8
After the third year:
$4000*1.05^3 + $4000*1.07^3 = $9530.672
$8000*1.06^3 = $9528.128
After the nth year:
The first option gives $4000*(1.05^n + 1.07^n)
The second option gives $8000*(1.06^n)
= $4000*(2)*(1.06^n)
= $4000*(1.06^n + 1.06^n)
Because 1.07^n - 1.06^n > 1.06^n - 1.05^n for n>1, the first option will be a better investment.
Tom walks 1/3 of a mile in 1/4 of an hour. At this rate, how many miles will tom walk in 1 hour ?
Tom walks at a rate of 4/3 miles per hour, which means he will walk 1.333 miles in one hour at this rate.
To find out how many miles Tom will walk in 1 hour, we need to determine the distance he covers in 1/4 of an hour and then calculate how much he would walk in 1 hour at that rate.
Given Tom walks 1/3 of a mile in 1/4 of an hour, first find how much he walks in 1 hour:
1/3 mile ÷ (1/4 hour) = 1/3 mile * 4/1 hour = 4/3 miles in 1 hour
Therefore, Tom will walk 4/3 miles in 1 hour at that pace.
Tom would walk 4/3 miles or 1 and 1/3 miles in 1 hour given that he walks 1/3 of a mile in 1/4 of an hour based on the unit rate calculation.
The question deals with finding out how many miles Tom will walk in 1 hour if he walks 1/3 of a mile in 1/4 of an hour. To calculate this, we use the concept of a unit rate, which is finding how much of something is done in one unit of something else, in this case, miles per hour. Since Tom walks 1/3 of a mile in 1/4 of an hour, we can set up a proportion to find out how many miles he would walk in 1 full hour.
The proportion is: (1/3) miles / (1/4) hour = x miles / 1 hour. To solve for x, we multiply both sides of the equation by 1 hour so x would equal (1/3) miles \/ (1/4) hour. To simplify the right side, we invert the fraction in the denominator and multiply:
x = (1/3) miles \/ (1/4). When we multiply by the reciprocal of 1/4, which is 4, we get x = (1/3) \/ 1 \/ 4 = 4/3 miles.
Therefore, at this rate, Tom would walk 4/3 miles or 1 and 1/3 miles in 1 hour.