Answer:
To graph y = 1/2x and y = x + 2, follow these steps:
Make a table of values for each equation. Choose several values of x, plug them into the equation, and solve for y. For y = 1/2x, you might choose x = -2, -1, 0, 1, and 2. For y = x + 2, you might choose the same values of x.
For y = 1/2x:
x | y
-2 | -1
-1 | -1/2
0 | 0
1 | 1/2
2 | 1
For y = x + 2:
x | y
-2 | 0
-1 | 1
0 | 2
1 | 3
2 | 4
Plot the points from each table on a graph. For y = 1/2x, plot the points (-2, -1), (-1, -1/2), (0, 0), (1, 1/2), and (2, 1). For y = x + 2, plot the points (-2, 0), (-1, 1), (0, 2), (1, 3), and (2, 4).
Draw a line through each set of points. The line for y = 1/2x should have a slope of 1/2 and pass through the point (0, 0). The line for y = x + 2 should have a slope of 1 and pass through the point (0, 2).
Label the axes and the lines. You can label the x-axis "x" and the y-axis "y". Label the line for y = 1/2x "y = 1/2x" and the line for y = x + 2 "y = x + 2".
Check your graph. Make sure each point is plotted correctly and the lines are drawn accurately. You can also check your graph by plugging in other values of x and making sure the corresponding points are on the lines.
Use simultaneously equation to eliminate one value and find the other then you find the value of the other one by substituting the value that u find from the other one eg: eliminate X find the value of y and then substitute y in one of your equation to find the value of X.
Solve the equation f(x) = 0
Factor the polynomial f(x) into linear factors.
State the multiplicity of each zero.
Use the table from your calculator to find any real zeros, then use synthetic division to find the remaining zeros. Show all steps.
f(x) = x ^ 4 + 2x ^ 3 + x ^ 2 + 8x - 12
The equation f(x) = 0 has four complex zeros: ±2i and 1 ± √2i, each with a multiplicity of 1.
To solve the equation f(x) = 0, we need to factor the polynomial f(x) into linear factors and then find the zeros of the equation.
Step 1: Factor the polynomial f(x) into linear factors.
f(x) = x ^ 4 + 2x ^ 3 + x ^ 2 + 8x - 12
= (x^2 + 4)(x^2 - 2x + 3)
Step 2: Find the zeros of the equation by setting each factor equal to zero and solving for x.
x^2 + 4 = 0
x^2 = -4
x = ±√(-4)
x = ±2i
x^2 - 2x + 3 = 0
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
x = (-(-2) ± √((-2)^2 - 4(1)(3)))/(2(1))
x = (2 ± √(-8))/2
x = (2 ± 2√2i)/2
x = 1 ± √2i
State the multiplicity of each zero.
The zeros of the equation are ±2i and 1 ± √2i. Each zero has a multiplicity of 1.
Use the table from your calculator to find any real zeros, then use synthetic division to find the remaining zeros.
There are no real zeros in this equation, as all of the zeros are complex numbers. Therefore, there is no need to use the table from the calculator or synthetic division to find the remaining zeros.
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Obtain the general solution to a 2nd order derivative for f(x)
with real coefficients if one of the roots of the auxiliary
equation is 3 + 7i
The general solution to a 2nd order derivative for f(x) with real coefficients can be obtained by finding the other root of the auxiliary equation and then using those roots to write the general solution.
Since one of the roots of the auxiliary equation is 3 + 7i, the other root must be the conjugate of this root, which is 3 - 7i. This is because the coefficients of the auxiliary equation are real, so the roots must come in conjugate pairs.
Now that we have both roots, we can write the general solution to the 2nd order derivative as:
f(x) = e^(3x)(C1*cos(7x) + C2*sin(7x))
where C1 and C2 are arbitrary constants.
This is the general solution to the 2nd order derivative for f(x) with real coefficients when one of the roots of the auxiliary equation is 3 + 7i.
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What fraction of each 100 chart is shaded?
The fraction when a 100-square grid has 70 shaded squares will be 7/10.
How to calculate the FractionA fraction simply means a piece of a whole. In this situation, the number is represented as a quotient such that the numerator and denominator are split. In this situation, in a simple fraction, the numerator as well as the denominator are both integers.
In this case, a 100-square grid has 70 shaded squares.
The fraction will be:
= Number it shaded square / Total square
= 70 / 100
= 7 / 10
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A 100-square grid has 70 shaded squares. Explain how you could express this model as a fraction.
Answer:
The fraction when a 100-square grid has 70 shaded squares will be 7/10.
June spent a quarter hour practicing her piano. Then she spent a half
hour cleaning her room, and an eighth hour putting away her clean
laundry. What fraction of an hour did June spend doing these chores? (this is for a kid i’m forced to tutor.)
Answer: 0.875 h or for a fraction 8 3/4
Step-by-step explanation:
June spent 15 minutes practicing her piano, 30 minutes cleaning her room and 7.5 minutes putting away her dirty laundry so a total of 52.5 minutes
Answer:
8 3/4
Step-by-step explanation: IK IM SORRY im sooo confusing!! And it makes me frustrated too...
Show all work to calculate the following exactly, include any reference triangle (a) For x > 0, if y = arccot(3x/8), find cos(y). (b) Show HOW to use your TI to approximate: sec^-1 (67). Write a sentence to tell Explain what the answer tells us. (c) Show how to find the exact value of cos^-1(cos(-5phi/6)). Show reference triangles for support:
(a) cos(y) = adjacent/hypotenuse = 3x/√(64 + 9x^2) b) The angle whose secant is 67 is approximately 89.149 degrees. C) The exact value of cos^-1(cos(-5phi/6)) is phi/3.
For x > 0, if y = arccot(3x/8), find cos(y).Solution: First, we draw the reference triangle with y = arccot(3x/8). Since y = arccot(3x/8), we have:
tan(y) = 8/3x. This means that the opposite side is 8 and the adjacent side is 3x.
Using the Pythagorean Theorem, we can find the hypotenuse: h = √(8^2 + (3x)^2) = √(64 + 9x^2). Now, we can find cos(y) using the definition of cosine: cos(y) = adjacent/hypotenuse = 3x/√(64 + 9x^2)
To find sec^-1 (67) using a TI calculator, we can use the inverse cosine function: sec^-1 (67) = cos^-1 (1/67) On the TI calculator, we can enter: cos^-1 (1/67). And the calculator will give us an approximate value of 89.149 degrees.
The angle whose secant is 67 is approximately 89.149 degrees.
To find the exact value of cos^-1(cos(-5phi/6)).
The reference triangle for 5phi/6 is a 30-60-90 triangle, with the hypotenuse equal to 2, the opposite side equal to √3, and the adjacent side equal to 1.
Therefore, we have: cos(5phi/6) = adjacent/hypotenuse = 1/2. Now, we can find the exact value of cos^-1(cos(-5phi/6)): cos^-1(cos(-5phi/6)) = cos^-1(1/2) = phi/3. Therefore, the exact value of cos^-1(cos(-5phi/6)) is phi/3.
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given is a circle whose equation is x^2 + y^2 - 4x + 6y -12 = 0.
find the tangent distance from (8,6).
A. 11.31
B. 8.45
C. 9.59
D. 23.11
In a circle whose equation is [tex]x^2 + y^2 - 4x + 6y -12 = 0[/tex]. The correct answer is option C. 9.59.
Equation of the circle in standard form by completing the square for both x and y terms:
[tex](x^2 - 4x) + (y^2 + 6y) = 12(x - 2)^2 + (y + 3)^2 = 12 + 4 + 9(x - 2)^2 + (y + 3)^2 = 25[/tex]
then Center: (2, -3) and Radius: 5
Equation of the line passing through the point (8,6) and the center of the circle (2,-3): Slope =
[tex](6 - (-3))/(8 - 2) = 9/6 = 3/2y - 6 = (3/2)(x - 8)y = (3/2)x - 6[/tex]
Substitute the equation of the line into the equation of the circle and solve for x:
[tex](x - 2)^2 + ((3/2)x - 6 - 3)^2 = 25(x - 2)^2 + ((3/2)x - 9)^2 = 25 (5/4)x^2 - (19/2)x + 80 = 0[/tex]
Use the quadratic formula to find the x-intercepts of the line and circle:
[tex]x = (-(-19/2) ± √((-19/2)^2 - 4(5/4)(80)))/(2(5/4))x = (19/2 ± √(361/4 - 400))/(5/2)x = (19/2 ± √(-39/4))/(5/2)x = (19 ± √(-39))/(5)[/tex]
Use the distance formula to find the distance between the point (8,6) and the x-intercepts:
[tex]Distance = √((x - 8)^2 + (y - 6)^2)Distance = √(((19 ± √(-39))/5 - 8)^2 + ((3/2)(19 ± √(-39))/5 - 6)^2)Distance = √((-19/5)^2 + (-39/5)^2)Distance = √(961/25 + 1521/25)Distance = √(2482/25)Distance = 9.59[/tex]
Therefore, the tangent distance from the point (8,6) to the circle is 9.59. The correct answer is option C
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Solve the problem. The width of a rectangle is 2 ft less than 4 times the length. Write a model for the width W in terms of the length L. Select one: a. W=4L - 2 b. W=4L+2 c.W=2L - 4 d. W = 2L +4
The width of a rectangle is 2 ft less than 4 times the length, so the model for the width W in terms of the length L is W = 4L - 2.
To solve this problem, we need to create a model for the width W in terms of the length L. According to the problem, the width of the rectangle is 2 ft less than 4 times the length. This can be written as:
W = 4L - 2
This equation represents the relationship between the width and the length of the rectangle. It shows that the width is equal to 4 times the length, minus 2. Therefore, the correct answer is a. W=4L - 2.
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A rectangular flag for a club has the following design on it. A rectangle of length 24 inches and width 9 inches has a flag inside with a length of 18 inches and width of 5 inches has a triangle is cut out from its right side and the remaining length of the flag is 12 inches. What is the area of the shaded region on this flag? A 75in2 B 90in2 C 141in2 D 216in2
Complete the rules for g (z) so that the graph represents it.
g(x) = -10, -15 ≤ x < -10
g(x) = , -10 ≤ x < -8
g(x) = 8, 10 ≤ x < 15
The rule for the function g(x) when completed is g(x) = -10, -15 ≤ x < -10; g(x) = -8, -10 ≤ x < -8; g(x) = -6, -8 ≤ x < -1; g(x) = 2, -1 ≤ x < 1; g(x) = 4, 1 ≤ x < 10; g(x) = 8, 10 ≤ x < 15
Completing the rule for the function g(x)Given
The graph of the function g(x) such that the function g(x) is a piecewise function and each sub-function is represented by horizontal lines
To complete the function definition, we write out the y value and the domain of the functions based on the current domain
Following the above statements, we have the following function definition for g(x)
g(x) = -10, -15 ≤ x < -10
g(x) = -8, -10 ≤ x < -8
g(x) = -6, -8 ≤ x < -1
g(x) = 2, -1 ≤ x < 1
g(x) = 4, 1 ≤ x < 10
g(x) = 8, 10 ≤ x < 15
The above is the definition of the function g(x)
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On a washington road map, Abigail measured a distance of 3.2 in from Snohomish to Seattle. The scale of this map is 1 inch = 10 miles what is the actual distance between Snohomish to Seattle?
Using the unitary method we found that the actual distance between Snohomish to Seattle is 32 miles.
What is meant by the unitary method?
The unitary approach is a strategy for problem-solving that involves first determining the value of a single unit, and then multiplying that value to determine the required value. We typically utilise this technique for maths computations. This method allows us to calculate both the value of many units from the value of one unit and the value of many units from the value of one unit.
Given,
The measured distance from Snohomish to Seattle on a road map = 3.2 inches
The scale of the map is given as:
1 inch = 10 miles
The ratio between a distance on a map and its actual distance on the ground is known as the map's scale. The use of scale is necessary for generating an accurate map and makes it simple to establish the real size of any area on the map.
The actual distance can be found using the unitary method.
So it is given 1 inch on the map = 10 miles on the road
Then 3.2 inches on the map = 10 * 3.2 = 32 miles on the road
Therefore using the unitary method we found that the actual distance between Snohomish to Seattle is 32 miles.
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Comment résoudre une équation ?
Ex: 3x + 4= 2x + 9
Given:-
[tex] \tt \: 3x + 4 = 2x + 9[/tex][tex] \: [/tex]
Solution:-
[tex] \tt \: 3x + 4 = 2x + 9[/tex][tex] \: [/tex]
[tex] \tt \: 3x - 2x = 9 - 4[/tex][tex] \: [/tex]
[tex] \tt \: 1x = 5[/tex][tex] \: [/tex]
[tex] \tt \: x = \cancel\frac{5}{1} [/tex][tex] \: [/tex]
[tex] \underline{ \underline{ \color{black}{ \tt \: x = 5}}}[/tex][tex] \: [/tex]
hope it helps!:)
Bonjour !
Il faut isoler x.
3x + 4 = 2x + 9
3x - 2x = 9 - 4
x = 5
A stock broker has found his investment window to be: a. sell if price are in the top 4% range and b. buy if prices are in the bottom 15% range. The price of amazon stocks average at 2540AED with a standard deviation of 150AED. On a given day, the price of the stock is 2900AED, what should be his course of action?
On a given day where the price of the stock is 2900AED, the stock broker should sell the stock.
The stock broker should sell the stock if the price is in the top 4% range. To determine if the price of 2900AED is in the top 4% range, we need to calculate the z-score and compare it to the z-score for the top 4% range.
The z-score formula is:
z = (x - μ) / σ
Where:
x = the value we are interested in (2900AED)
μ = the mean (2540AED)
σ = the standard deviation (150AED)
Plugging in the values, we get:
z = (2900 - 2540) / 150
z = 360 / 150
z = 2.4
The z-score for the top 4% range is 1.75. Since the z-score of 2.4 is greater than 1.75, the price of 2900AED is in the top 4% range. Hence, he should sell the stocks.
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need help you all with this question:(
Answer:
True both each.
Step-by-step explanation:
Your teacher would probably like you to remember the Trig Identities or allow you to have a cheat sheet for the Quiz or Test.
[tex]\frac{sin(\alpha )}{sin^{2}(\alpha) +cos2\alpha } \\\\\frac{sin(\alpha )}{sin^{2}(\alpha) +[cosx^{2} (\alpha)-sinx^{2} (\alpha )]} \\\\\frac{sin(\alpha )}{sin^{2}(\alpha) +cosx^{2} (\alpha)-sinx^{2} (\alpha )}\\\\\frac{sin(\alpha )}{[sin^{2}(\alpha) +cosx^{2} (\alpha)]-sinx^{2} (\alpha )}[/tex]
[tex]\frac{sin(\alpha )}{1-sinx^{2} (\alpha )}\\\\\frac{sin(\alpha )}{cos^{2} (\alpha )}\\\\\frac{sin(\alpha )}{cos^{2} (\alpha )}\\\\\frac{sin(\alpha )}{cos(\alpha )*cos(\alpha )}\\\\\frac{sin(\alpha )}{cos(\alpha )}*\frac{1}{cos(\alpha )} \\\\tan(\alpha )*sec(\alpha )\\\\\frac{1}{cot(\alpha )}*sec(\alpha )\\\\\frac{sec(\alpha )}{cot(\alpha )}[/tex]
ctIONS through a given point with a given slope ope -(2)/(3) passing through the point (3,4)
The equation of the line with slope -(2)/(3) passing through the point (3, 4) is y = -(2)/(3)x + 6. To find the equation of a line that passes through a given point with a given slope, we can use the point-slope formula. The point-slope formula is: y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point that the line passes through. In this case, the slope is -(2)/(3) and the point is (3, 4).
Plugging these values into the point-slope formula, we get:
y - 4 = -(2)/(3)(x - 3)
We can then rearrange the equation to get it into slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept:
y - 4 = -(2)/(3)x + 2
y = -(2)/(3)x + 6
So, the equation of the line with slope -(2)/(3) passing through the point (3, 4) is y = -(2)/(3)x + 6.
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- 3x + 3y = 6
x + 3y = 18
solve using substitution
find x and y
Answer:x = 3 and y = 5
Step-by-step explanation:
We have two equations:
-3x + 3y = 6 ........(1)
x + 3y = 18 ........(2)
We can use the second equation to solve for y in terms of x:
-4x = -12
x = 4
Now we substitute this expression for y into the second equation:
3 + 3y = 18
3y = 15
y = 5
Therefore, the solution to the system of equations is x = 3 and y = 5.
equation 1: - 3x + 3y = 6
equation 2: x + 3y = 18
start with equation 2:
x + 3y = 18
subtract 3y from both sides:
x = 18 - 3y
plug the new equation into equation 1:
- 3x + 3y = 6
- 3 (18 - 3y) + 3y = 6
multiply:
-54 + 9y + 3y = 6
collect like terms:
-54 + 12y = 6
add 54 to both sides, then divide the whole equation by 12:
12y = 60
y = 5
plug the y value into either equation to solve for x. for example, here is equation 2:
x + 3y = 18
x + 3(5) = 18
x + 15 = 18
x = 3
check answer:
x + 3y = 18
3 + 3(5) = 18
3 + 15 = 18
18 = 18 is true
Exercise 2.4.28 LetAbe ann×nmatrix and letIbe then×nidentity matrix. a. IfA2=0, verify that(I−A)−1=I+A. b. IfA3=0, verify that(I−A)−1=I+A+A2.
a. For A2 = 0, it is proved that (I - A)-1 = I + A.
b. For A3 = 0, (I - A)-1 = I + A + A2.
a. If A2 = 0, then (I - A)-1 = I + A
Proof:
(I - A)-1 = (I + A)-1
(I + A)-1 = (I + A)(I - A)-1
(I + A)(I - A)-1 = I + A
Therefore, (I - A)-1 = I + A
b. If A3 = 0 then (I - A)-1 = I + A + A2
Proof:
(I - A)-1 = (I + A + A2)-1
(I + A + A2)-1 = (I + A + A2)(I - A)-1
(I + A + A2)(I - A)-1 = I + A + A2
Therefore, (I - A)-1 = I + A + A2
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How does lowering interest rates by a government’s central bank affect the economy?
Lowering interest rate by a government's central bank can have a significant impact on the economy, as it affects the cost of borrowing money for businesses and individuals, as well as the return on savings and investments.
Lowering interest rate reduces the cost of borrowing money, making it easier for businesses and individuals to access credit. It can also encourage consumers to spend more money, as they may have more disposable income due to lower borrowing costs or higher investment returns.
It can also lead to increased inflation, as businesses and individuals have more money to spend and demand for goods and services increases and reduced returns on savings and investments, which can discourage people from saving and lead to more spending
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Solve the following application question.
You must clearly show your variable designation(s), equation(s), and algebraic solution to the equation(s) to receive full credit.
Shiro bought some meat for a barbeque. The beef costs $6.00 per pound and the chicken costs $4.50 per pound. He bought a total of 18 pounds of meat and spent $96. How much of each type of meat did Shiro purchase?
Shiro purchased 11 pounds of beef and 4 pounds of chicken.
How to solveShiro purchased a total of 18 pounds of meat, spending $96.
We can designate the beef as x pounds, and the chicken as y pounds. To find the solution, we set up two equations:
6x + 4.5y = 96 x + y = 18
To solve this system of equations, we can subtract 6x from both sides of the first equation, giving us 4.5y = 96 - 6x.
We can substitute this expression into the second equation, giving us x + (96 - 6x) = 18.
We can then simplify this equation, giving us 7x = 78, and thus x = 11.
Now we can substitute x = 11 into the original equation, 6x + 4.5y = 96, to find y = 4.
Therefore, Shiro purchased 11 pounds of beef and 4 pounds of chicken.
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In the triangle ABC.
angle ABC is twice the size of angle BAC.
angle ACD is 31° more than angle ABC.
Work out the size of angle ACB.
The size of angle ACB is approximately 50.66°.
What is the angle sum property?The angle sum property of a triangle states that the sum of the interior angles of a triangle is 180 degrees.
We are given that;
Angle ABC= Angle ACD + 31°
Now,
Since angle ABC is twice angle BAC, we can write:
angle BAC = x
angle ABC = 2x
angle ACB = 180° - (angle BAC + angle ABC)
Substituting these values into the equation for angle ACD, we get:
angle ACD = angle ABC + 31° = 2x + 31°
Since angles ACB and ACD form a straight line, we can write:
angle ACB + angle ACD = 180°
Substituting the values we know, we get:
(angle BAC + angle ABC) + (angle ABC + 31°) = 180°
3x + 31° = 180°
3x = 149°
x = 49.67°
Therefore, the angle of triangles answer will be 50.66°.
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Grayson bought snacks for his team's practice. He bought a bag of popcorn for $3.50 and a 5-pack of juice bottles. The total cost before tax was $11.05. Write and solve an equation which can be used to determine
Answer:
7.55
Step-by-step explanation:
3.50 + x = 11.05, x = 11.05 - 3.50, x = 7.55
Answer:
Step-by-step explanation:
Let [tex]x[/tex] be the cost of a juice bottle. Then
[tex]3.50+5x=11.05[/tex]
Solve this:
[tex]5x=7.55[/tex] (subtracted 3.50 from both sides)
[tex]x=7.55/5=1.51[/tex] (divided both sides by 5)
So juice bottles cost $1.51
A cube of length 8cm is enlarged with a scale factor of 1 1/2
find the length of the enlargement
The length of the enlargement of the cube, given the scale factor and the length, would be 12 cm.
How to find the length of the enlargement ?When a shape is enlarged by a scale factor of k, all its dimensions are multiplied by k. In this case, the scale factor is 1 1/2, which can be written as 3/2 in fraction form. Therefore, the length of the enlargement is:
Length of enlargement = Scale factor x Original length
Length of enlargement = (3/2) x 8 cm
Length of enlargement = 12 cm
So, the length of the enlargement is 12 cm.
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PLEASE HELP ME OUT THIS IS SO DIFFICULT
What is the measurement of the longest line segment in a right rectangular prism that is 12.6 centimeters long, 3.2 centimeters wide, and 7 centimeters tall
The right rectangular prism's longest line segment measures around 14.79 cm in length.
The right rectangular prism's longest line segment can be found in this way?We are aware that the rectangular prism's body diagonal makes up the longest line segment inside a right rectangular prism. The longest distance is that between points A and D. AD is the longest line segment as a result. To determine a line segment's length, use a scale. Straight line segments can be measured with it.
The Pythagorean theorem can be used to determine how long the space diagonal is:
a = length = 12.6 cm
b = width = 3.2 cm
c = height = 7 cm
You may determine the space diagonal (d) by using:
d = sqrt(a² + b² + c²)
Substituting the values, we get:
d = sqrt(12.6² + 3.2² + 7²) cm
d = sqrt(159.76 + 10.24 + 49) cm
d = sqrt(219) cm
d ≈ 14.79 cm
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Question:
What is the measurement of the longest line segment in a right rectangular prism that is 12.6 centimeters long, 3.2 centimeters wide, and 7 centimeters tall, and that connects two vertices that are not on the same face of the prism?
Using the formula for finding the simple interest, I = Prt, find the Interest earned in a savings account by depositing $9,800 for 15 months at 5% simple interest
let's recall that a year has 12 months, thus 15 months is really 15/12 of a year.
[tex]~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\dotfill & \$9800\\ r=rate\to 5\%\to \frac{5}{100}\dotfill &0.05\\ t=years\to \frac{15}{12}\dotfill &\frac{5}{4} \end{cases} \\\\\\ I = (9800)(0.05)(\frac{5}{4}) \implies I = 612.5[/tex]
640 times 36 equals ?? with step by step instructions on how to solve.
Answer:
To multiply 640 by 36, we use the standard multiplication algorithm:
640
x 36
------
3840 (6 times 640)
+25600 (3 times 640, shifted one digit to the left)
-------
23040
Therefore, 640 times 36 equals 23,040.
Complete the identity
The identity cos²θ/2 is (1 + cosθ)/2
What is Trigonometry?Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.
We can use the half-angle identity for the cosine function to solve this problem:
cos²θ/2 = (1 + cosθ)/2
Therefore, to find cos²θ/2 , we need to know the value of cosθ.
If the value of cosθ is known, we can substitute it into the half-angle identity to find cos²θ/2
Hence, the identity cos²θ/2 is (1 + cosθ)/2
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Leveled Practice In 8-11, complete each statement.
8. A die has 12 sides shown as follows: 6 triangles,
3 circles, and 3 squares.
The probability rolling a triangle is
Q
2
out of 12, or
or
%.
(s
The probability of rolling a triangle is 50%
How to determine the probability of rolling a triangleFrom the question, we have the following parameters that can be used in our computation:
Sides = 12
triangles = 6
Circles = 3
Squares = 3
The die has a total of 6 + 3 + 3 = 12 sides.
Since there are 6 triangles on the die, the probability of rolling a triangle is:
P(triangle) = number of favorable outcomes / total number of outcomes
Substitute the known values in the above equation, so, we have the following representation
P(triangle) = 6 / 12
Evaluate
P(triangle) = 50%
Hence, the probability of rolling a triangle on the die is 1/2 or 0.5.
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4) If ABCD is a parallelogram, Find the value of Angle B. ∟AB=(6x-15)°
∟CD=(4x-11)°
5.) PQRS is a parallelogram, Find the value of angle R ∟QR=(8x - 12)° ∟PS=(3x + 5)º 7) Given the following quadrilateral is a rhombus, find the measure of x ∟AB=23°
∟BC=y°
∟CD=x°
x = 23°
4) To find the value of Angle B, we can use the alternate interior angles theorem, which states that if two parallel lines are cut by a transversal, the alternate interior angles are equal. In this case, we can say that ∟AB=∟CD, since ABCD is a parallelogram. Therefore, we can set 6x-15 = 4x-11 and solve for x to get x = 4. Plugging this back in, we can find that ∟AB = 6(4)-15 = 15°.
5) To find the value of Angle R, we can use the same method as above. We can set ∟QR=∟PS, since PQRS is a parallelogram. Therefore, we can set 8x-12 = 3x+5 and solve for x to get x = 7. Plugging this back in, we can find that ∟QR = 8(7)-12 = 56°.
7) To find the measure of x, we can use the fact that the quadrilateral is a rhombus. A rhombus is a quadrilateral with all four sides equal in length, so we can say that ∟AB = ∟CD. Therefore, we can set 23=x and solve for x to get x = 23°.
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Find general solutions of the following trigono metric equations. a. \( 4 \sin ^{2}(x)-4 \sin (x)+1=0 \) b. \( \tan (x) \sin (x)+\sin (x)=0 \) c. \( 3 \csc ^{2}(\theta)=4 \) d. \( 2 \sin (3 x)-1=0 \)
The general solutions for this equation are [tex]\( x = \frac{\pi}{18} + \frac{2n\pi}{3} \)[/tex] and [tex]\( x = \frac{5\pi}{18} + \frac{2n\pi}{3} \)[/tex].
The general solutions of the given trigonometric equations can be found by applying the basic trigonometric identities and solving for the unknown variable.
a. [tex]\( 4 \sin ^{2}(x)-4 \sin (x)+1=0 \)[/tex]
This equation can be solved by using the quadratic formula. Let \( u = \sin (x) \), then the equation becomes [tex]\( 4u^2 - 4u + 1 = 0 \)[/tex]. Using the quadratic formula, we get:
[tex]\( u = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(4)(1)}}{2(4)} \)\\\( u = \frac{4 \pm \sqrt{16 - 16}}{8} \)\\\( u = \frac{4}{8} = \frac{1}{2} \)[/tex]
Now, we can substitute back \( u = \sin (x) \) and solve for x:
[tex]\( \sin (x) = \frac{1}{2} \)\\\( x = \arcsin (\frac{1}{2}) \)\\\( x = \frac{\pi}{6} + 2n\pi \) or \( x = \frac{5\pi}{6} + 2n\pi \)[/tex], where n is an integer.
The general solutions for this equation are [tex]\( x = \frac{\pi}{6} + 2n\pi \)[/tex] and[tex]\( x = \frac{5\pi}{6} + 2n\pi \)[/tex].
b. [tex]\( \tan (x) \sin (x)+\sin (x)=0 \)[/tex]
This equation can be solved by factoring out \( \sin (x) \):
[tex]\( \sin (x)(\tan (x) + 1) = 0 \)[/tex]
This equation will be true if either[tex]\( \sin (x) = 0 \) or \( \tan (x) + 1 = 0 \)[/tex].
For \( \sin (x) = 0 \), the general solutions are[tex]\( x = n\pi \)[/tex] where n is an integer.
For \( \tan (x) + 1 = 0 \), the general solutions are [tex]\( x = \frac{3\pi}{4} + n\pi \)[/tex], where n is an integer.
The general solutions for this equation are [tex]\( x = n\pi \) and \( x = \frac{3\pi}{4} + n\pi \).[/tex]
c. [tex]\( 3 \csc ^{2}(\theta)=4 \)[/tex]
This equation can be solved by isolating [tex]\( \csc ^{2}(\theta) \)[/tex] and taking the square root of both sides:
[tex]\( \csc ^{2}(\theta) = \frac{4}{3} \)[/tex]
[tex]\( \csc (\theta) = \pm \sqrt{\frac{4}{3}} \)[/tex]
Now, we can use the identity[tex]\( \csc (\theta) = \frac{1}{\sin (\theta)} \)[/tex]to solve for \( \theta \):
[tex]\( \frac{1}{\sin (\theta)} = \pm \sqrt{\frac{4}{3}} \)\( \sin (\theta) = \pm \sqrt{\frac{3}{4}} \)[/tex]
The general solutions for this equation are[tex]\( \theta = \arcsin (\pm \sqrt{\frac{3}{4}}) + 2n\pi \)[/tex], where n is an integer.
d. [tex]\( 2 \sin (3 x)-1=0 \)[/tex]
This equation can be solved by isolating[tex]\( \sin (3x) \)[/tex] and taking the inverse sine of both sides:
[tex]\( \sin (3x) = \frac{1}{2} \)\( 3x = \arcsin (\frac{1}{2}) \)\\\( 3x = \frac{\pi}{6} + 2n\pi \) or \( 3x = \frac{5\pi}{6} + 2n\pi \)[/tex], where n is an integer.
The general solutions for this equation are [tex]\( x = \frac{\pi}{18} + \frac{2n\pi}{3} \)[/tex] and [tex]\( x = \frac{5\pi}{18} + \frac{2n\pi}{3} \)[/tex].
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What is the minimum possible parameter of a rectangle whose length and breadth are whole numbers and area is 60 sq. Units?
The minimum possible parameter of the rectangle is 34 units.
What is a rectangle?
A rectangle is a quadrilateral having four sides and the sum of the angles is 180 in the rectangle the opposite two sides are equal and parallel and the two sides are at 90-degree angles.
Let's assume the length of the rectangle is L and the width is W. The area of the rectangle is given as 60 sq. units.
Area of rectangle = Length × Width = L × W = 60
We are looking for the minimum perimeter of the rectangle. Perimeter of rectangle = 2(L + W)
To find the minimum perimeter, we need to find the minimum values of L and W that satisfy the condition that the area is 60.
The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
If we choose L = 1 and W = 60, then the area is 1 × 60 = 60.
If we choose L = 2 and W = 30, then the area is 2 × 30 = 60.
If we choose L = 3 and W = 20, then the area is 3 × 20 = 60.
If we choose L = 4 and W = 15, then the area is 4 × 15 = 60.
If we choose L = 5 and W = 12, then the area is 5 × 12 = 60.
If we choose L = 6 and W = 10, then the area is 6 × 10 = 60.
The minimum perimeter occurs when L and W are the closest in value, which is achieved when L = 5 and W = 12. Thus, the minimum perimeter of the rectangle is:
Perimeter = 2(L + W) = 2(5 + 12) = 34
Therefore, the minimum possible parameter of the rectangle is 34 units.
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