From the system of equations one apple pie costs $6 and one lemon meringue pie costs $7.
What exactly is an equation system?A group of equations that need to be solved concurrently is referred to as a system of equations. The values of the variables that satisfy each and every one of the system's equations are the solutions to the corresponding equations. Systems of equations can be solved using a variety of techniques, such as substitution, elimination, and graphing. Systems of equations can be used to simulate situations in business, science, and engineering when there are several variables at play. Systems of differential equations, linear equations, and quadratic equations are a few examples of systems of equations.
Let us suppose the cost of the apple pie = x.
Let us suppose the cost of the meringue pie = y.
Given that, Dan sold 14 apple pies and 9 lemon meringue pies for a total of $147.
14x + 9y = 147
For Jenny:
8x + 12y = 132
Using the elimination method we have:
Multiplying Dan's equation by 4 gives:
56x + 36y = 588
Multiplying Jenny's equation by 3 gives:
24x + 36y = 396
Subtracting the second equation from the first:
32x = 192
x = 6
Substitute the value of x in equation:
14(6) + 9y = 147
84 + 9y = 147
9y = 63
y = 7
Hence, from the system of equations one apple pie costs $6 and one lemon meringue pie costs $7.
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Does the table below describe a linear or an exponential function?
A. The x-values are equally spaced, and the y-values change by equal differences, so the function is exponential.
B. The x-values are equally spaced, and the y-values change by equal factors, so the function is exponential.
C. The x-values are equally spaced, and the y-values change by equal differences, so the function is linear.
D. The x-values are equally spaced, and the y-values change by equal factors, so the function is linear.
Answer:
C) The x-values are equally spaced, and the y-values change by equal differences, so the function is linear.-----------------------
As per table we see:
x-values increase by 1;y-values increase by 2.Since the difference of both variables is common, the function is linear.
The matching choice is C.
A new automobile is purchased for $20,000. If V = 20,000(0. 8)x, gives the car’s value after x years, about how long will it take for the car to be worth $8,200?
It will take about 5.18 years for the car to be worth 8,200.
We can use the given formula to find the value of the car after x years:
[tex]V = 20,000(0.8)^x[/tex]
We want to know how long it will take for the car to be worth 8,200. So we can set V equal to 8,200 and solve for x:
[tex]8,200 = 20,000(0.8)^x[/tex]
Dividing both sides by 20,000 gives:
[tex]0.41 = (0.8)^x[/tex]
Taking the logarithm of both sides with base 0.8, we get:
[tex]log0.8(0.41) = x[/tex]
Using a calculator, we can find:
x ≈ 5.18
Therefore, it will take about 5.18 years for the car to be worth 8,200.
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Consider thatzis a real number, determine the value (s) forβsuch thatz=1+iββ+2iβ=±2β=±2β=±1β=±3None of the given options
The given equation is z = 1 + iβ/β + 2i. We need to determine the value(s) for β such that the equation holds true.
First, we can multiply both sides of the equation by β + 2i to get rid of the denominator:
z(β + 2i) = 1 + iβ
Next, we can distribute the z on the left-hand side of the equation:
zβ + 2iz = 1 + iβ
Now we can rearrange the equation to get all the β terms on one side:
zβ - iβ = 1 - 2iz
We can factor out the β on the left-hand side:
β(z - i) = 1 - 2iz
Finally, we can divide both sides of the equation by z - i to get the value of β:
β = (1 - 2iz)/(z - i)
This is the value of β that satisfies the given equation. None of the given options match this value, so the correct answer is "None of the given options."
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In Southern California, there is a six-mile section of Interstate 5 that decreases 2,500 feet in elevation as it descends Grapevine Hill in the Tejon Pass. What is the angle of descent?
angle = tan⁻¹(2,500 / 31,680)
angle ≈ 4.51 degrees
Therefore, the angle of descent is approximately 4.51 degrees.
What does a math angle mean?When two rays collide at a given point, an angle is created. Indicated by the symbol is the "angle," also known as the "opening" between these two beams. Many angles, such as 60°, 90°, etc., are usually stated as numbers in degrees.
To find the angle of descent, we can use the tangent function, which relates the opposite and adjacent sides of a right triangle:
tan(angle) = opposite / adjacent
In this case, the opposite side is the change in elevation, which is 2,500 feet, and the adjacent side is the distance traveled, which is 6 miles or 31,680 feet (since 1 mile = 5,280 feet).
So we have:
tan(angle) = 2,500 / 31,680
To solve for the angle, we can take the inverse tangent (or arctangent) of both sides:
angle = tan⁻¹(2,500 / 31,680)
Using a calculator, we get:
angle ≈ 4.51 degrees
Therefore, the angle of descent is approximately 4.51 degrees.
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A primary school enrolls 50 kindergarteners every year. Next year, they are moving to a smaller building and will only be able to enroll 42 kindergarteners. To the nearest percent, what is the percent of decrease in the amount of students who will be enrolled?
Answer:
16 percent decrease
Step-by-step explanation:
A food merchant thinks that a new marketing campaign will will increase sales of the product in supermarkets by an average of 50 units per week. For a sample of 20 supermarkets, the mean increase in sales was 41.3 units with a standard deviation of 12.2 units. Contrast, at the 5% level, the null hypothesis that the population mean of the increase in sales is at least 50 units, indicating any assumptions made Use both critical value approach and pvalue approach. No excel
a)1.746
b)0.119
To compare the null hypothesis that the population mean of the increase in sales is at least 50 units against the observed mean of 41.3 units, we will use both the critical value approach and the p-value approach.
For the critical value approach, we need to use the standard deviation of 12.2 units as well as the sample size of 20 supermarkets. This gives us a critical value of 1.746. As the observed mean (41.3 units) is less than 1.746 standard deviations away from the mean of 50 units, we can conclude that the null hypothesis cannot be rejected at the 5% level of significance.
For the p-value approach, we will use the same sample size and standard deviation. This gives us a p-value of 0.119, which is greater than the 5% level of significance. Thus, we can also conclude that the null hypothesis cannot be rejected at the 5% level of significance.
It is important to note that we are assuming that the data is normally distributed, and that there is no bias in the sample.
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Please please help me! This is due ASAP
sin 200 degrees cos 80 degrees- cos 200 degrees sin 80 degrees
Thank you!!
Answer: -sqrt(3)/4
Step-by-step explanation:
To solve this trigonometric expression, we can use the following trigonometric identities:
sin (180 + x) = -sin x
cos (180 + x) = -cos x
Using these identities, we can rewrite the expression as:
sin(200°)cos(80°) - cos(200°)sin(80°)
= sin(180° + 20°)cos(80°) - cos(180° + 20°)sin(80°)
= -sin(20°)cos(80°) + cos(20°)sin(80°)
= (cos(20°) - sin(20°))sin(80°)
= sin(70°)sin(80°)
= (cos(20°) - cos(150°))/2
= (-sqrt(3)/2 - sqrt(3)/2)/2
= -sqrt(3)/4
Therefore, sin(200°)cos(80°) - cos(200°)sin(80°) = -sqrt(3)/4.
Evaluate. Write your answer as a fraction or whole number without exponents. 9^-1
Answer:
1/9
Step-by-step explanation:
I plugged it into a calculator.
Anita brings 7 dolls to her grandma’s house. These dolls represent 20% of Anita’s doll collection. What is the total number of dolls in the collection? And WHY?
Answer:
35
Step-by-step explanation:
(7/20) *100=35
This is why because if 20 percent is equal to 7, so 40 % will be 14, 80% 28 and 100% will be 35
Answer: 35 dolls
Step-by-step explanation:
7/x and 20/100
20 times 5 is 100
and 7 times 5 is 35. Therefore 35 dolls are in Anita’s collection
Solve for x. Round to the nearest tenth, if necessary.
Answer:
x = 50
Step-by-step explanation:
sin 40º = 32 / x
0.64 = 32 / x
x = 32 / 0.64
x =~ 50
An image of a rhombus is shown.
a rhombus with a base of 20 centimeters and a height of 17 centimeters
What is the area of the rhombus?
148 cm^2
74 cm^2
340 cm^2
170 cm^2
What is rhombus ?
A rhombus is a special type of parallelogram in which two pairs of opposite sides are congruent. That means all the sides of a rhombus are equal.
What is the Area of a Rhombus?
The area of a rhombus can be defined as the amount of space enclosed by a rhombus in a two-dimensional space. To recall, a rhombus is a type of quadrilateral projected on a two dimensional (2D) plane, having four sides that are equal in length and are congruent.
Using Diagonals A = ½ × d1 × d2
Using Base and Height A = b × h
Using Trigonometry A = b2 × Sin(a)
the correct answer is the option (C) 340 cm^2
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Which equation finds the value of x in the triangle below?
Right triangle X Y Z is shown. Side X Z has a length of 12, side Z Y has a length of x, and hypotenuse X Y has a length of z. The angle opposite to side x is 20 degrees.
Cotangent 70 degrees = StartFraction x Over 12 EndFraction
Cotangent 70 degrees = StartFraction 12 Over x EndFraction
Secant 20 degrees = StartFraction x Over 12 EndFraction
Secant 20 degrees = StartFraction 12 Over x EndFraction
we have that
1) cot 70=x/12
Cot 70=[opposite side]/[adjacent side]=x/12
The option 1 is correct
2) cot 70=12/x
Cot 70=[opposite side]/[adjacent side]=x/12
The option 2 is not correct
3)sec 20=x/12
Sec 20 =1/cos20
Cos 20==[opposite side]/[hypotenuse]=12/z
1/(1/cos20)-------> 1/(12/z)------> z/12
The option 3 is not correct
4) sec 20=12/x
Sec 20 =1/cos20
Cos 20==[opposite side]/[hypotenuse]=12/z
1/(1/cos20)-------> 1/(12/z)------> z/12
The option 4 is not correct
the correct value is
1) cot 70=x/12
5/x+3-3/x-3 = 5x/x2-9 A. write the value or values of the
variable that make a denominator zero. x= __ B. what is the
solution of the equation? what is the solution set
The solution set of the equation is {0}
A. The values of the variable that make a denominator zero are [tex]x = 3[/tex] and [tex]x = -3[/tex]. This is because when [tex]x = 3[/tex] or [tex]x = -3[/tex], the denominator of the fraction [tex]x^2 - 9[/tex] becomes zero, causing the entire equation to become undefined.
B. To find the solution of the equation, we can first multiply both sides by the common denominator, [tex]x^2 - 9[/tex]:
[tex](5/x+3-3/x-3)(x^2-9) = (5x/x^2-9)(x^2-9)[/tex]
Simplifying and combining like terms, we get:
[tex]5x - 9 - 3x + 9 = 5x[/tex]
[tex]2x = 5x[/tex]
Subtracting [tex]5x[/tex] from both sides, we get:
[tex]-3x = 0[/tex]
Dividing by -3, we find that the solution of the equation is x = 0.
Therefore, the solution set of the equation is {0}.
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If a line passes through the points (1,2) and (−0.5,1), what is the distance between the x and y intercepts of that line? (A) √13/6 (B) √13/2 (C) 2√13/3 (D) 3√13/2 (E) None of these
The distance between the x and y intercepts of that line is √13/2. The correct answer is (B).
To find the distance between the x and y intercepts of a line, we can use the distance formula:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
First, we need to find the x and y intercepts of the line. The x intercept is where the line crosses the x-axis, so we can set y to 0 and solve for x. Similarly, the y intercept is where the line crosses the y-axis, so we can set x to 0 and solve for y.
Using the point-slope form of a line, we can find the equation of the line passing through the points (1,2) and (−0.5,1):
y - 1 = (1 - 2)/(−0.5 - 1)(x - (-0.5))
y - 1 = -2(x + 0.5)
y = -2x - 1
Now we can find the x and y intercepts:
For the x intercept, set y to 0:
0 = -2x - 1
2x = -1
x = -1/2
For the y intercept, set x to 0:
y = -2(0) - 1
y = -1
So the x intercept is (-1/2, 0) and the y intercept is (0, -1). Now we can use the distance formula to find the distance between these two points:
d = √((-1/2 - 0)² + (0 - (-1))²)
d = √((-1/2)² + (1)²)
d = √(1/4 + 1)
d = √(5/4)
d = √13/2
Therefore, the distance between the x and y intercepts of the line is √13/2.
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CEREAL A company claims that its boxes of cereal contain 19.1 ounces of
cereal. The actual amount of cereal in the box can be within 0.5 ounce of the
advertised amount. What is the range of possible amounts of cereal c in a box?
25. Find the scale factor of JKLM to EFGH.
26. List all pairs of congruent angles.
27. Write the ratios of the corresponding side lengths in a statement of proportionality
28. Find the values of x, y, and z.
29. Find the perimeter of each polygon
30. Find the ratio of the perimeters of JKLM to EFGH
31. Find the area of each polygon
32. Find the ratio of the areas of JKLM to EFGH
25. The scale factor is 2.5
26. The congruent angles include
∠ E = ∠ J = 65 degrees
∠ F = ∠ K = 90 degrees
∠ G = L = 90 degrees
∠ H = ∠ M = 115 degrees
How to find the scale factor25. The scale factor of JKLM and EFGH is calculated using corresponding sides
side EF * scale factor = side JK
8 * scale factor = 20
scale factor = 20 / 8
scale factor = 2.5
26. The congruent angles include
∠ E = ∠ J = 65 degrees
∠ F = ∠ K = 90 degrees
∠ G = L = 90 degrees
∠ H = ∠ M = 360 - 90 - 90 - 65 = 115 degrees
27. the ratios of the corresponding side lengths
JK / EF is proportional to KL / FG is proportional to LM / GH is proportional to MJ / EH
28. the values of x, y, and z
x = 2.5 * FG = 2.5 * 11 = 27.5
y = 30 / 2.5 = 12
z = 65 degrees
29. the perimeter of each polygon
the smaller polygon EFGH = 8 + 11 + 3 + 12 = 34
the bigger polygon JKLM = 20 + 27.5 + (3 * 2.5) + 30 = 85
30. the ratio of the perimeters of JKLM to EFGH
= 85 / 34 = 2.5
31. the area of each polygon
the smaller polygon EFGH = 0.5(3 + 8) * 11 = 60.5
the bigger polygon JKLM = 0.5(20 + 7.5) * 27.5 = 378.125
32. the ratio of the areas of JKLM to EFGH
= 378.125 / 60.5 = 6.25
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Could someone please help me with part b pleaseeeee its 60% of my grade
We can see here that the experimental probability that it will snow in March is:
Fraction: 2/5Percentage: 40%.What is experimental probability?Experimental probability is the probability that an event will occur based on actual experiments or observations.
It is calculated by performing an experiment or conducting observations, counting the number of times the event of interest occurs, and dividing that count by the total number of trials or observations.
Mathematically, the formula for the experimental probability is defined by:
Probability of an Event P(E) = Number of times an event occurs / Total number of trials.
Number of times it snowed = 2
Total number of trials = 5.
Thus, the experimental probability will be: 2/5.
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Please Answer!! Thank you!!!
Answer:
isosceles triangle.
Step-by-step explanation:
To classify triangle WXY by its sides, we need to determine whether all three sides are equal, two sides are equal, or all three sides are different. We can do this by calculating the lengths of the sides using the distance formula:
Side WX:
sqrt[(x2 - x1)^2 + (y2 - y1)^2] = sqrt[(-1 - 4)^2 + (-3 - (-10))^2] = sqrt[5^2 + 7^2] = sqrt(74)
Side WY:
sqrt[(x2 - x1)^2 + (y2 - y1)^2] = sqrt[(11 - 4)^2 + (-5 - (-10))^2] = sqrt[7^2 + 5^2] = sqrt(74)
Side XY:
sqrt[(x2 - x1)^2 + (y2 - y1)^2] = sqrt[(11 - (-1))^2 + (-5 - (-3))^2] = sqrt[12^2 + 2^2] = sqrt(148)
Since the lengths of sides WX and WY are equal, but the length of side XY is different, we can classify triangle WXY as an isosceles triangle. Specifically, it is an isosceles triangle with sides WX and WY equal.
what is the nth term a_(n) of the geometric sequence with a_(1)=4 r=(1)/(2) and n=5
1/4
The nth term of a geometric sequence can be found using the formula:
a_(n) = a_(1) * r^(n-1)
where a_(1) is the first term of the sequence and r is the common ratio.
Given a_(1) = 4, r = 1/2, and n = 5, we can substitute these values into the formula to get:
a_(5) = 4 * (1/2)^(5-1)
= 4 * (1/2)^4
= 4 * (1/16)
= 1/4
Therefore, the 5th term of the geometric sequence is a_(5) = 1/4.
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In a certain investigation: 460 persons were involved in the study: and based on an enquiry on their age. it was known that 75% of them were 22 or more years. The following frequency distribution shows the age composition of the persons under study. Find the mean and modal life of condensers_ Mid; age in years No. of persons 18 f; 23 28 33 90 122 ; 38 56 48 24 20 33'
Therefore , the solution of the given problem of percentage comes out to be 38 years old is the median age.
What does a percentage actually mean?In statistics, a "a%" is a figure or statistic that is expressed as a percentage of 100. The words "pct," "pct," but instead "pc" are also not frequently used. However, the sign "%" is frequently used to represent it. The percentage sum is flat; there are no dimensions. Percentages are truly integers because their numerator almost always equals 100. Either the % symbol (%) or the additional term "fraction" must come before a number to denote that it is a percentage.
Here,
The following algorithm must be used to determine the mean age:
=> mean = (all years added together) / (total number of persons)
The representative age for each age group is the age that falls in the middle of the range. With this approach, we can determine the total age as follows:
=>18(9) + 23(28) + 28(33) + 33(90) + 38(122) + 56(48) + 48(24) + 76(20) + 1089(0.25)
=> 162 + 644 + 924 + 2970 + 4636 + 2688 + 1152 + 1520 + 272.25
=>18(9) + 23(28) + 28(33) + 33(90) + 38(122) + 56(48) + 48(24) + 76(20) + 1089(0.25)
According to the information provided, there are 460 people in total.
=> mean = 17745.25 / 460
=> mean = 17745.25 / 460
As a result, the average lifespan is roughly 38.552 years.
The age category in this instance with the highest frequency has a midpoint of 38 and a frequency of 122. As a result, 38 years old is the median age.
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Which of the following are the coordinates of point B' , the image of point B after a translation of (x-4, y-3)?
Answer: 1, -2
Step-by-step explanation:
Answer:
(1,-2)
Step-by-step explanation:
x-4 you move it to the left 4 units and y-3 you move it down 3 units to get (1,-2)
POLYNOMIALS AND FACTORING Introduction to the GCF of two monomials Find the greatest common factor of 8m^(2) and 7b^(3).
The greatest common factor of 8m2 and 7b3 is the largest monomial that can divide both 8m2 and 7b3. So, the greatest common factor of 8m^(2) and 7b^(3) is 1.
The greatest common factor (GCF) of two monomials is the product of the greatest common factor of their coefficients and the greatest common factor of their variables. In this case, the greatest common factor of the coefficients is 1, since 8 and 7 have no common factors other than 1. The GCF of the variables is 1, since m and b have no common factors. Therefore, the GCF of 8m^(2) and 7b^(3) is 1*1 = 1.
Here is a step-by-step explanation:
1. Find the GCF of the coefficients: GCF(8,7) = 1
2. Find the GCF of the variables: GCF(m^(2),b^(3)) = 1
3. Multiply the GCF of the coefficients and the GCF of the variables: 1*1 = 1
4. The GCF of 8m^(2) and 7b^(3) is 1.
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12. R(x) = (2x+6)(x-1) (4-2)(x+2) Find any x-intercept(s) and any y-intercepts(s)
The x-intercepts of R(x) are (-3,0), (1,0), and (-2,0), and the y-intercept is (0,-24).
To find the x-intercepts of R(x) = (2x+6)(x-1)(4-2)(x+2), we need to set R(x) equal to 0 and solve for x:
0 = (2x+6)(x-1)(4-2)(x+2)
This equation can be simplified to:
0 = (2x+6)(x-1)(2)(x+2)
We can use the zero product property to find the x-intercepts:
2x+6 = 0, x-1 = 0, 2 = 0, x+2 = 0
Solving for x, we get:
x = -3, x = 1, x = -2
So, the x-intercepts are (-3,0), (1,0), and (-2,0).
To find the y-intercepts, we need to set x equal to 0 and solve for R(x):
R(0) = (2(0)+6)(0-1)(4-2)(0+2)
R(0) = (6)(-1)(2)(2)
R(0) = -24
So, the y-intercept is (0,-24).
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Consider the following.
Given:
△XYZ with XY ≅ XZ ≅ YZ
ZW ⊥ XY with W on XY
YZ = 8
Find:ZW
Create a drawing as needed.
Since
XY ≅ XZ ≅ YZ, △XYZ
is ---Select--- an equilateral a scalene triangle. This means △XYZ is also ---Select--- equiangular right obtuse .
Find the measures of the following angles in degrees.
m∠WYZ= °m∠YWZ= °m∠YZW= °
Find WY.
WY =
Find ZW. Give your answer in both simplest radical form and as an approximation correct to two decimal places.
simplest radical formZW=
approximationZW=
ZW = 5.656
Since XY ≅ XZ ≅ YZ, △XYZ is an equilateral triangle. This means △XYZ is also equiangular.
Find the measures of the following angles in degrees:
m∠WYZ = 60°
m∠YWZ = 60°
m∠YZW = 60°
Find WY:
WY = 8
Find ZW: Give your answer in both simplest radical form and as an approximation correct to two decimal places.
simplest radical form ZW = 4√2
approximation ZW = 5.656854249492381
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A store buys 10 sweaters for $50 and sells them for $390. If p represents the total profit in dollars and cents when the store buys any number of sweaters, s, write a proportional equation for p in terms of s that matches the context.
The proportional equation for p in terms of s which matches the given context is p = 34s.
What is a proportional equation?
A proportional equation is written as y = kx. The equation's variables are the letters y and x. The proportionality constant, which never changes, is symbolised by the letter k.
We are given that a store buys 10 sweaters for $50 and sells them for $390.
Cost of one sweater = $50 / 10 = $5
Sale price of one sweater = $390 / 10 = $39
We know that total profit = Sale price - Cost price.
So, we get
p = 39s - 5s
p = 34s
Hence, the proportional equation for p in terms of s which matches the given context is p = 34s.
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Please help will give brainilest
Step-by-step explanation:
The first step is to add 6 to both sides
3x =15+6
3x=21
Divide both sides by 3
X = 7
how do i solve it on paper
Answer: Less than three would be like 2 or 1 or any negative number.
Step-by-step explanation: Three munis anything is less than the 3 you had to start with.
Sami spent $250 to buy a domain name for her graphic design website. She has nine years before she needs to renew or sell the domain name. What will the domains amortized expense be after one year
A. 17.30
B. 24.20
C. 27.80
D. 32.20
The domains amortized expense be after one year is $27.80, the correct option is C
What is Algebra?Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.
The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.
We are given that;
The expenditure of sami= $250
Now,
To calculate the amortized expense of the domain name, we need to divide the total cost by the number of years before it needs to be renewed or sold. In this case, Sami spent $250 and has nine years before renewal, so the annual amortized expense will be:
$250 / 9 = $27.78
Rounding this to the nearest cent
Therefore, by algebra the answer will be $27.78.
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Evaluate:
8
power of 8/3
The expression when evaluated is 256
How evaluate the expressionFrom the question, we have the following parameters that can be used in our computation:
8 power of 8/3
Mathematically, this can be expressed as
8^(8/3)
When evaluated using a calculator, we have
8^(8/3) = 256
Hence, the solution is 256
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A cart weighing 40 lb is placed on a ramp inclined at 15° to the horizontal. The cart is held in place by a rope inclined at 60° to the horizontal, as shown in the figure. Find the force that the rope must exert on the cart to keep it from rolling down the ramp.
Answer:
11.97 lb
Step-by-step explanation:
To find the force that the rope must exert on the cart to keep it from rolling down the ramp, we need to resolve the forces acting on the cart along the direction of the ramp and perpendicular to the ramp.
First, we resolve the weight of the cart into its components. The weight of the cart acting vertically downwards can be resolved into a component perpendicular to the ramp and a component parallel to the ramp.
The component perpendicular to the ramp is given by:
W_perp = W * cos(theta) = 40lb * cos(15°) = 38.6lb
The component parallel to the ramp is given by:
W_parallel = W * sin(theta) = 40lb * sin(15°) = 10.4lb
where W is the weight of the cart, and theta is the angle of inclination of the ramp.
Next, we resolve the force exerted by the rope into its components. The force exerted by the rope can be resolved into a component perpendicular to the ramp and a component parallel to the ramp.
The component perpendicular to the ramp is given by:
F_perp = F * cos(phi) = F * cos(60°) = 0.5F
The component parallel to the ramp is given by:
F_parallel = F * sin(phi) = F * sin(60°) = 0.87F
where F is the force exerted by the rope, and phi is the angle of inclination of the rope.
To keep the cart from rolling down the ramp, the force exerted by the rope must balance the weight of the cart along the direction of the ramp. That is,
F_parallel = W_parallel
0.87F = 10.4lb
Solving for F, we get:
F = 11.97lb
Therefore, the force that the rope must exert on the cart to keep it from rolling down the ramp is approximately 11.97lb.