Step-by-step explanation:
2 bakers make 30 loaves in 1 hour.
that means each baker just by him-/herself can make 15 loaves in 1 hour.
so,
1 baker's "speed" = 15 loaves/hr
15 l/h = 30l/x
x = 30/15 = 2 hours (1 baker takes 2 hours to make 30).
3 bakers' speed = 3×15/hr = 45l/h
45l/h = 30l/x
x = 30/45 = 2/3 hours (3 bakers need 2/3 hours = 40 minutes to make 30 loaves).
5 bakers = 5×15l/h = 75 l/h.
75l/h = 30l/x
x = 30/75 = 2/5 hours (5 bakers need 2/5 hours = 2×60/5 = 24 minutes to make 30 loaves).
6 bakers = 6×15l/h = 90 l/h.
90 l/h = 30/x
x = 30/90 = 1/3 hours (6 bakers need 1/3 hour = 60/3 = 20 minutes to make 30 loaves).
15 bakers = 15×15 l/h = 225 l/h
225 l/h = 30/x
x = 30/225 = 2/15 hours (15 bakers need 2/15 hours = 2×60/15 = 2×4 = 8 minutes to make 30 loaves).
Determine the inverse of the matrix C equals a matrix with 2 rows and 2 columns. Row 1 is 9 comma 7, and row 2 is 8 comma 6..
The inverse matrix of C is equal to a matrix with 2 rows and 2 columns. Row 1 is 3 comma negative 3.5, and row 2 is negative 4 comma 4.5.
The inverse matrix of C is equal to a matrix with 2 rows and 2 columns. Row 1 is negative 3 comma 3.5, and row 2 is 4 comma negative 4.5.
The inverse matrix of C is equal to a matrix with 2 rows and 2 columns. Row 1 is 6 comma 8, and row 2 is 7 comma 9.
The inverse matrix of C is equal to a matrix with 2 rows and 2 columns. Row 1 is negative 9 comma 8, and row 2 is 7 comma negative 6.
Answer: [-4.5 -3.5]
[-4 -3]
Step-by-step explanation:
Music and a matrix calculator online, mwah
Find the missing length.
The value of the missing side length is 9.
What is the value of the missing length?The figures in the image are two similar triangle.
Side lengths of smaller triangle are:
x and (10-4)
Side lengths of larger triangle are:
(x+6) and 10
Since the triangle are similar, we take the proportion of the side lengths and solve for x.
x/(10-4) = (x+6)/10
x/6 = (x+6)/10
Cross multiply
x × 10 = 6( x + 6 )
Apply distributive property to eliminate the parenthesis.
x × 10 = 6×x + 6×6
10x = 6x + 36
10x - 6x = 36
4x = 36
x = 36/4
x = 9
Therefore, the value of x is 9.
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A tabletop in the shape of a trapezoid has an area of 6,350 square centimeters. Its longer base measures 115 centimeters, and the shorter base is 85 centimeters. What is the height?
The height of the trapezoid can be found by dividing the area (6,350 cm2) by the average of the two bases (100 cm). The height is 63.5 cm.
The formula used to find the area of a trapezoid is A = (1/2)(b1 + b2)h, where b1 and b2 are the lengths of the two bases and h is the height. We are given the area (A) and the lengths of both bases (b1 and b2). All we need to do is solve for h.
To do this, we first need to find the average of the two bases. The average of the two bases is the sum of the two bases divided by two. In this case, the longer base (b1) is 115 cm and the shorter base (b2) is 85 cm, so we can calculate the average as (115 + 85) ÷ 2 = 100 cm.
We can now substitute this value into the area formula and solve for h. A = (1/2)(100)h, so h = A ÷ (1/2)(100). Plugging in the area given (6,350 cm2), we get h = 6,350 ÷ (1/2)(100) = 63.5 cm. The height of the trapezoid is 63.5 cm.
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USE PEDMAS PLEASE 30 POINTS
Select the expression that makes the equation true.
one half x (3 x 5 + 1) – 2 = ___
4 x (2 + 3)
(4 x 3) ÷ 2
6 ÷ 3 + 2
6 + 8 ÷ 4
The expression that makes the equation true= 4 x (2 + 3).
What are functions?A relation is any subset of a Cartesian product.
As an illustration, a subset of is referred to as a "binary connection from A to B," and more specifically, a "relation on A."
A binary relation from A to B is made up of these ordered pairs (a,b), where the first component is from A and the second component is from B.
Every item in a set X is connected to one item in a different set Y through a connection known as a function (possibly the same set).
A function is only represented by a graph, which is a collection of all ordered pairs (x, f (x)).
Every function, as you can see from these definitions, is a relation from X.
Hence, The expression that makes the equation true=
4 x (2 + 3).
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Answer:
a
Step-by-step explanation:
none
Jamie is laying stone pavers along his patio and walkway. The walkway is 3.5 feet wide. the patio measures 7 feet from top bottom and 8 feet accross. how much area does jamie need to cover.
84 sq ft area does jamie need to cover.
Describe Area?
The region enclosed by an object's form is referred to as the area. The area of the form is the area that the figure or any other two-dimensional geometric shape occupies in a plane.
A two-dimensional shape's area, which is expressed in square units, is the total surface area it may occupy. The square metre (m2), a derived measure, serves as the area unit in the SI system. On a sheet of paper, draw a square using a pencil. It has two dimensions. Its Area refers to the area that the form occupies on the paper.
the area of the walkway is:
Area of walkway = length x width = 8 ft x 3.5 ft = 28 sq ft
Next, let's calculate the area of the patio. The patio measures 7 feet from top to bottom and 8 feet across. Therefore, the area of the patio is:
Area of patio = length x width = 7 ft x 8 ft = 56 sq ft
Now, to find the total area that Desiree needs to cover, we just add the area of the walkway and the area of the patio:
Total area = area of walkway + area of patio
Total area = 28 sq ft + 56 sq ft
Total area = 84 sq ft
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The exponential function below is used by a biologist and her lab tomorrow the growth of some bacteria. A=2e^0.3t in the function the value of a depends on the value of t what statement describes that relationship 
The relationship between A and t is exponential, meaning that as time increases, the number of bacteria grows at an accelerating rate.
What is exponential function?An exponential function is a mathematical function of the form:
f(x) = a^x
where "a" is a positive constant called the base, and "x" is the independent variable. The base "a" is usually a number greater than 1, but it can also be a fraction between 0 and 1.
Exponential functions have some key properties that make them useful for modeling many natural phenomena. One of the most important properties is that they exhibit exponential growth or decay, meaning that the rate of change of the function is proportional to the current value of the function.
In the given question,
In the exponential function A=2e^(0.3t), the value of A depends on the value of t. Specifically, A is a function of t, where t is the independent variable and A is the dependent variable. This means that as the value of t changes, the value of A changes as well.
The function describes the growth of some bacteria, where A represents the number of bacteria at time t. The value of t represents the time elapsed since the start of the observation period. The function suggests that the growth rate of the bacteria is proportional to the current number of bacteria, with a growth rate of 0.3 per unit time.
The relationship between A and t is exponential, meaning that as time increases, the number of bacteria grows at an accelerating rate. This relationship can be observed by plotting the function on a graph, where A is plotted on the y-axis and t is plotted on the x-axis. The resulting curve will be a steeply upward-sloping curve that gets steeper and steeper as time goes on.
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Thelma and David built a recycling bin. The area of the base is 72 ft² and it is 14 feet high.
What is the volume of the recycling bin?
Answer: The volume of the recycling bin can be calculated by multiplying the area of the base by the height of the bin.
Given that the area of the base is 72 ft² and the height is 14 feet, we can use the formula:
Volume = Area of Base × Height
Substituting the given values, we get:
Volume = 72 ft² × 14 feet
Volume = 1008 cubic feet
Therefore, the volume of the recycling bin is 1008 cubic feet.
Step-by-step explanation:
Which of the following equations will produce the graph shown below?
A. X^2- y^2/4= 1
B. Y^2/9 - x^2/4=1
C. Y^2- x^2/9= 1
D. Y^2/2 - x^2/4= 1
From following equations option B will produce the hyperbola graph shown in the figure
what is hyperbola ?
A hyperbola is a type of conic section, which is a curve that is formed by the intersection of a plane and a double cone. A hyperbola can also be defined as the set of all points in a plane, the difference of whose distances from two fixed points (called the foci) is a constant.
In the given question,
Based on the shape of the hyperbola shown on the y-axis graph, we can tell that the hyperbola has a vertical transverse axis, which means that its equation must have the form:
(y - k)² / a² - (x - h)² / b² = 1
where (h, k) is the center of the hyperbola, a is the distance from the center to the vertices, and b is the distance from the center to the co-vertices.
Option A is not correct because it produces a hyperbola with a horizontal transverse axis, whereas the given graph has a hyperbola with a vertical transverse axis.
We can eliminate option D since its equation has a transverse axis that is not vertical.
Next, we can eliminate option A since the coefficient of x² is positive, which means that the transverse axis is horizontal.
Option C has a transverse axis that is also horizontal, so we can eliminate it as well.
That leaves us with option B, which has a vertical transverse axis and its equation fits the form we determined earlier. Therefore, the equation Y²/9 - x²/4=1 will produce the hyperbola shown on the y-axis graph
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suppose that a friend is helping put on a fundraiser for the local animal shelter. one activity is a game using a bowl that contains six green marbles and eight blue marbles. to play the game, each person draws two marbles without replacement and without looking. if both marbles are green, the player wins $25. if not, the player must donate $10 to the animal shelter. the marbles are then replaced for the next player. calculate the expected value of the game for the player.
Answer:
To calculate the expected value of the game for the player, we need to find the probability of each possible outcome and the corresponding payoff or cost, and then multiply each probability by its payoff or cost, and sum the results. Let's first find the probability of each possible outcome:
- Probability of drawing two green marbles: (6/14) * (5/13) = 0.164
- Probability of not drawing two green marbles: 1 - 0.164 = 0.836
If both marbles are green, the player wins $25, so the payoff is $25. If not, the player must donate $10 to the animal shelter, so the cost is -$10. Therefore, the expected value of the game for the player is:
Expected value = (Probability of winning * Payoff) + (Probability of losing * Cost)
Expected value = (0.164 * $25) + (0.836 * -$10)
Expected value = $4.10 - $8.36
Expected value = -$4.26
The expected value of the game for the player is -$4.26, which means that on average, the player can expect to lose $4.26 per game. Therefore, playing this game is not a good bet for the player.
What is the solution to this equation?
x/5 = 25
A. x=20
B. x=30
C. x=20
D. x=125
Answer:D. x=125
Step-by-step explanation:
Answer:D
Step-by-step explanation:
125/5= 25
Help me find area
9th grade
Answer:
Step-by-step explanation: add all of them together
Please help me I really stuck
Answer:
We can find the value of Y when k = 5/4 by substituting k = 5/4 in the given expression for Y:
Y = 16 × 10^8 × k
Y = 16 × 10^8 × (5/4)
Y = 20 × 10^8
To express this in standard form, we can convert it to scientific notation:
Y = 2.0 × 10^9
Step-by-step explanation:
y^(5/4) = (16×10^(8k))^(5/4)
remember all the rules of exponents :
an exponent of an exponent : we multiply both exponents.
x^(a/b) =
[tex] \sqrt[b]{ {x}^{a} } = ({ \sqrt[b]{x} })^{a} [/tex]
and
(a×b)^c = a^c × b^c
so, with that we can do the trick here :
y^(5/4) = (16×10^(8k))^(5/4) =
= 16^(5/4) × 10^(8k × 5/4)
16^(5/4) =
[tex] \sqrt[4]{ {16}^{5} } = ( \sqrt[4]{16} )^{5} = ( \sqrt[4]{ {2}^{4} })^{5} = {2}^{5} = 32[/tex]
10^(8k × 5/4) = 10^(40k/4) = 10^(10k)
so, the result is
32 × 10^(10k)
[tex]32 \times {10}^{10k} [/tex]
make a cylinder and find
CSA
TSA
Volume
Answer:
This means that a cylinder has two kinds of surface areas -Total Surface Area (TSA) and Curved Surface Area (CSA). For a cylinder whose base radius is 'r' and height is 'h': TSA of cylinder = 2πr2 + 2πrh (or) 2πr (r + h) CSA of cylinder = 2πrh.
You are studying an amoeba through a microscope. The amoeba moves on a grid-indexed microscope slide in a straight line from square B3 to square G7. the side length of each grid square is 2 mm. How far does the amoeba travel? round your answer to the nearest 10th.
To find the distance traveled by the amoeba, we need to calculate the Euclidean distance between the starting point B3 and the ending point G7.
Using the Pythagorean theorem, we can find the length of the diagonal line connecting these two points:
d = √[(G - B)² + (7 - 3)²] * 2
where G - B represents the horizontal distance between the two points in number of squares.
In this case, G - B = 6 (from B to G there are 6 squares horizontally).
Substituting the values, we get:
d = √[6² + 4²] * 2 = √(36 + 16) * 2 = √52 * 2 ≈ 11.4 mm
Therefore, the amoeba travels approximately 11.4 mm. Rounded to the nearest 10th, the distance is 11.4 ≈ 11.4 mm.
A car loses its value at a rate of 27% each year. How long will it take for its value to halve?
It will take nearly three years for the car's value to be half of its original price.
What can be submitted for bands in a single -arm row?
Resistance bands, loop bands, tube bands with handles, and cable machines can be used for single-arm rows
In a single-arm row exercise, a variety of equipment can be used to add resistance and challenge the muscles. Here are some examples of what can be submitted for bands in a single-arm row
Resistance bands: Resistance bands are a popular option for single-arm rows. They come in different colors to indicate the level of resistance and can be easily adjusted to increase or decrease the difficulty. To perform a single-arm row with a resistance band, step on the center of the band with one foot and grasp the other end with one hand
Loop bands: Loop bands are a type of resistance band that is formed into a loop. They can be placed around the wrists or forearms to provide resistance during a single-arm row. To perform a single-arm row with a loop band, loop the band around your hand and hold onto the other end with your other hand.
Tube bands with handles: Tube bands with handles are another option for single-arm rows. These bands have handles on each end, allowing for a more comfortable grip. To perform a single-arm row with a tube band, attach one end of the band to a stationary object and hold onto the other end with one hand.
Cable machines: Cable machines are a piece of gym equipment that can also be used for single-arm rows. To perform a single-arm row on a cable machine, adjust the weight to the desired resistance and attach a handle to the cable.
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for her phone service, jenny pays a monthly fee of $24, and she pays an additional $0.06 per minute of use. the least she has been charged in a month is .what are the possible numbers of minutes she has used her phone in a month?use for the number of minutes, and solve your inequality for $96.36.
Jenny must have used at least 1600 minutes of her phone service to incur a minimum monthly fee of $96.36. This can be calculated by solving the inequality $24 + 0.06x ≥ $96.36 where x is the number of minutes.
Rearranging this equation to 0.06x ≥ $72.36 and solving for x gives us the result x ≥ 1600.
Given that, For her phone service, Jenny pays a monthly fee of $24, and she pays an additional $0.06 per minute of use. The least she has been charged in a month is $96.36.To find the possible numbers of minutes she has used her phone in a month.
Inequality for $96.36 is(0.06m + 24) ≥ 96.36where m is the number of minutes used. In order to solve the above inequality, we will simplify it first(0.06m + 24) ≥ 96.360.06m + 24 - 24 ≥ 96.36 - 24.060.06m ≥ 72.36m ≥ 72.36/0.06m ≥ 1206So, the possible numbers of minutes she has used her phone in a month is greater than or equal to 1206 minutes. Answer: $\boxed{m≥1206}$
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What is the similarity between these numbers: 7, 21, 69, 71?
A They are all multiples of 7
B They are all factors of 71
C They are all prime numbers
D They have a pattern of + 14
Answer:
A
Step-by-step explanation:
The answer is A because the numbers only go under category A
A circle with radius of \greenD{1\,\text{cm}}1cmstart color #1fab54, 1, start text, c, m, end text, end color #1fab54 sits inside a \blueD{3\,\text{cm} \times 4\,\text{cm}}3cm×4cmstart color #11accd, 3, start text, c, m, end text, times, 4, start text, c, m, end text, end color #11accd rectangle. What is the area of the shaded region? Round your final answer to the nearest hundredth
Given the rectangle's dimensions, its area is 12 cm² and the circle's area is π cm². Thus, the shaded region's area is about 8.57 cm².
The area of the shaded region is equal to the area of the rectangle minus the area of the circle.
The area of the rectangle is given by:
Area of rectangle = length x width = 4 cm x 3 cm = 12 cm²
The area of the circle is given by:
Area of circle = π x radius² = π x (1 cm)² = π cm²
Therefore, the area of the shaded region is:
Area of shaded region = Area of rectangle - Area of circle
Area of shaded region = 12 cm² - π cm²
Area of shaded region ≈ 8.57 cm² (rounded to the nearest hundredth)
Hence, the area of the shaded region is approximately 8.57 cm².
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the missing figure is in the image attached below
I just want to know what to put in the number line
Answer:
270 students walked to school
30% students walked to school
Step-by-step explanation:
270 /900 is 30%
Answer:
Step-by-step explanation:
first mark the 30% line. then divide 900 by 0.3 to get 30% of the students (270)
similar to 3.5.31 in rogawski/adams. find f(40)(x) if f(x)=x−4 by first finding the general solution. (use symbolic notation and fractions where needed.)
The general solution of function f^(40)(x) = 0 if f(x)=x−4.
The general solution of a function is an expression that includes all possible solutions to a given differential equation or equation. It usually contains arbitrary constants that are determined by applying initial or boundary conditions.
The general solution allows for a range of solutions, rather than a single specific solution. This allows for flexibility in solving problems, as different initial or boundary conditions can result in different specific solutions.
We first find the general solution of f(x) by integrating the function 40 times.
f(x) = x - 4
f'(x) = 1
f''(x) = 0
f'''(x) = 0
f''''(x) = 0
and so on, until we get to:
f^39(x) = 0
Now, we can take the 40th derivative of the function to get the final answer:
f^(40)(x) = 0
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which of the following graph/s pass the vertical line test and would it be considered a function
Applying the concept of vertical line test, only graph 1 is qualified to be a function.
What is vertical line test?An analytical tool for determining if a given relation is a function is the vertical line test. The test is drawing a vertical line somewhere on the relation's graph and determining if it crosses the graph several times.
Every vertical line must cross the graph at exactly one place in order for the relation to be a function. This is due to the fact that there is no ambiguity in the mapping between inputs and outputs, where each input (x-value) corresponds to a certain output (y-value).
On the other hand, the connection is not a function if the vertical line crosses the graph more than once. This is due to the unclear mapping between inputs and outputs and the presence of at least two outputs that correspond to the same input.
Without having to explicitly analyze the relation for all potential inputs, the vertical line test is a helpful technique for rapidly identifying whether a particular relation is a function or not. It is especially beneficial for relational graphs like scatterplots, line graphs, and curves.
Using vertical line test, only graph 1 is a function
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4. Which expression would be used to find the sum of the products in the area model
below?
20
A 600+600 + 40 + 4
B 600+600 + 400
30
2
2
C600 +60 + 400 + 40
600 +60 + 40 + 4
600+60 + 400 + 4
Answer:
The expression that would be used to find the sum of the products in the area model would be:
C) 600 + 60 + 400 + 40 = 1100
Write the absolute value equations in the form |x|=c (where b is a number and c can be either a number an expression) that has the following solution set.
Two solutions x=1/2, x=-1/3
According to the given information, the absolute value is |x - 7/6| = 5/6.
What is an algebraic expression?
A mathematical expression known as an algebraic expression can include variables, integers, and mathematical operations including addition, subtraction, multiplication, and division.
When expressing quantities or values that might change depending on the values given to the expression's variables, algebraic expressions are utilised. Expressions in algebra can include one or more variables and be simple or complex.
Two solutions x=1/2, x=-1/3
|x - 7/6| = 5/6.
We know that the absolute value of (1/2 - 7/6) equals the absolute value of (-1/3 - 7/6) because they are opposites and have the same absolute value. So we need an expression that equals 5/6 when either 1/2 - 7/6 or -1/3 - 7/6 is plugged in for x.
To get an expression that equals 5/6 when 1/2 - 7/6 is plugged in for x, we can subtract 7/6 from 1/2 and take the absolute value of the result:
|1/2 - 7/6| = |-1/6| = 1/6.
To get an expression that equals 5/6 when -1/3 - 7/6 is plugged in for x, we can subtract 7/6 from -1/3 and take the absolute value of the result:
|-1/3 - 7/6| = |-9/6| = 3/2.
Since both expressions have the same absolute value of 5/6, we can use either one as the absolute value expression in the equation. Thus,
|x - 7/6| = 5/6.
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REFLECTION HELP If P= (-2,4), then find: Ry=1 (P)
(?, ?)
Answer:
P'(-2, -2)
Step-by-step explanation:
You want to know the location of P(-2, 4) after it is reflected in y=1.
ReflectionThe image point will be as far below the line y=1 as point P is above the line. The x-coordinate is unchanged.
P is 4 -1 = 3 units above the line y=1.
P' will be 3 units below the line y=1, so its y-coordinate is 1 -3 = -2.
The image point is P'(-2, -2).
__
Additional comment
Reflection in the line y=k is described by the transformation ...
(x, y) ⇒ (x, 2k-y)
P(-2, 4) ⇒ P'(-2, 2(1) -4) = P'(-2, -2)
if there are 10 students and sum of their age is 150 then find the mean age of students
[tex]10[/tex] pupils mean in age from sum [tex]150[/tex] to [tex]15[/tex] years.
What does the math mean?The sum of all numbers divided by the entire number of values determines the mean (also known as the arithmetic mean, which differs from the scaling factor) of a dataset. This indicator of central tendency is usually referred to as the "average".
How is the mean determined?Just dividing the total number of values inside a data collection by the sum of all of the values yields it. Both raw data and data that have been compiled into a frequency distribution table may be utilized in the computation. Often, average is referred to as mean or mathematical mean. Mean is only a way of summarizing the sample's average.
Mean age [tex]=[/tex] (sum of ages) / (number of students)
Mean age [tex]= 150 / 10[/tex]
Mean age [tex]= 15[/tex]
Therefore, the mean age of the [tex]10[/tex] students is [tex]15[/tex] years.
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Calculator
3.05 Quiz: Volumes of Cones
What is the approximate volume of a cone with a height of 12 in. and radius of 9 in.?
Use 3.14 to approximate pi, and express your final answer to the nearest hundredth.
Enter your answer as a decimal in the box.
in³
Rounding to the nearest hundredth, the approximate volume of the cone is 1017.36 cubic inches.
How do we find volume of cone ?To find the volume of a cone, we use the following formula:
V = 1/3 * π * r² * h
where V is the volume of the cone, π (pi) is a mathematical constant approximately equal to 3.14, r is the radius of the circular base of the cone, and h is the height of the cone.
To use the formula, we simply substitute the given values for r and h into the formula and simplify. Make sure that the radius and height are measured in the same units. The resulting volume will be in cubic units.
The formula for the volume of a cone is:
V = 1/3 * π * r² * h
where π is pi (approximately 3.14), r is the radius of the base of the cone, and h is the height of the cone.
Substituting the given values, we get:
V = 1/3 * 3.14 * 9² * 12
= 1/3 * 3.14 * 81 * 12
= 3.14 * 324
= 1017.36
Rounding to the nearest hundredth, the approximate volume of the cone is 1017.36 cubic inches.
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Mr Khan put $6000 in his bank. He withdraws it all four years later.
How much does he withdraw if the simple interest rate was 4% per year?
Step-by-step explanation:
If the simple interest rate is 4% per year, then the interest earned each year is calculated by multiplying the principal amount by the interest rate.
The interest earned in one year is:
Interest = Principal x Rate = $6000 x 0.04 = $240
After four years, the total interest earned is:
Total Interest = Interest x Time = $240 x 4 = $960
Therefore, the total amount Mr. Khan can withdraw after four years is:
Withdrawal Amount = Principal + Total Interest = $6000 + $960 = $6960
Answer:
If the simple interest rate was 4% per year, we can use the formula for simple interest to calculate the amount of interest Mr. Khan earned on his $6000 deposit:
Simple interest = P * r * t
Where P is the principal (the initial amount deposited), r is the interest rate (as a decimal), and t is the time (in years).
In this case, P = $6000, r = 0.04, and t = 4. Substituting these values into the formula, we get:
Simple interest = $6000 * 0.04 * 4 = $960
The total amount Mr. Khan can withdraw after four years is the sum of his initial deposit and the interest earned:
Total amount = Principal + Interest
Total amount = $6000 + $960 = $6960
Therefore, Mr. Khan can withdraw $6960 from his bank account after four years if the simple interest rate was 4% per year.
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x = 30° is a zero for y = tan 3(x +30°). True or False with justification
Answer: true
Step-by-step explanation:
true
Help me with these polar coordinate problems please!
I filled some in, could I get those checked? For the rest, please explain how I can solve them. I have formulas to use, but I don't know how to use them. They are:
[tex]x=rcos(\theta)\\y=rsin(\theta)\\x^2+y^2=\theta^2\\tan(\theta)=\frac{y}{x}[/tex]
The sooner the better please, because I have a quiz soon. I'll put this at a good amount of points so if you don't have a real answer...please don't answer. Thank you so much.
Convert the rectangular equation to polar form
3x-y+2=0
To convert the rectangular equation 3x - y + 2 = 0 to polar form, we can substitute x = r cos(theta) and y = r sin(theta), where r is the radius and theta is the angle in polar coordinates. This gives:
3(r cos(theta)) - (r sin(theta)) + 2 = 0
Simplifying the equation, we get:
r(3 cos(theta) - sin(theta)) = -2
Dividing both sides by the expression in the parentheses:
r = -2 / (3 cos(theta) - sin(theta))
This is the polar form of the equation.
Convert the rectangular equation to polar form
12.3x-y+2=0
13. xy=4
14. (x+y)-9(x-2)=0 72-36=0
15. y²-8x-16-0 (4-4)(3-4)=0
For the rectangular equation xy = 4, we can convert it to polar form using the substitution x = r cos(theta) and y = r sin(theta), which gives:
r cos(theta) * r sin(theta) = 4
r^2 sin(theta) cos(theta) = 4
Using the identity 2 sin(theta) cos(theta) = sin(2theta), we can simplify the equation to:
r^2 sin(2theta) = 8
r = sqrt(8 / sin(2theta))
This is the polar form of the equation.
For the rectangular equation (x+y) - 9(x-2) = 0, we can simplify it to:
x - 8y + 18 = 0
Then, we can convert it to polar form using the substitution x = r cos(theta) and y = r sin(theta), which gives:
r cos(theta) - 8r sin(theta) + 18 = 0
Simplifying the equation, we get:
r = 18 / (cos(theta) - 8sin(theta))
This is the polar form of the equation.
For the rectangular equation y^2 - 8x - 16 = 0, we can complete the square to get:
y^2 - 8x - 16 = (y - 0)^2 - 16 - 8x
Simplifying the equation, we get:
(y - 0)^2 = 8x + 16
Using the substitution x = r cos(theta) and y = r sin(theta), we get:
r^2 sin^2(theta) = 8r cos(theta) + 16
r^2 sin^2(theta) - 8r cos(theta) - 16 = 0
This equation does not simplify nicely into a standard form of polar equation, but it is still a valid polar form.
16. r = 4 sin θ
17.θ= (π/6)
18. r² = sin 2θ
19. r = 6/(2-3 sin θ)
To convert the polar equation r = 4 sin(theta) to rectangular form, we can use the following trigonometric identities:
sin(theta) = y / r
cos(theta) = x / r
Substituting these into the polar equation, we get:
r = 4 sin(theta)
r = 4 y / r
r^2 = 4 y
x^2 + y^2 = 4 y
This is the rectangular form of the equation.
The equation theta = pi/6 represents a line at an angle of pi/6 radians (30 degrees) from the positive x-axis in the polar coordinate system. In rectangular coordinates, this line has the equation y = x tan(pi/6) = x/sqrt(3).
To convert the polar equation r^2 = sin(2theta) to rectangular form, we can use the following trigonometric identities:
sin(2theta) = 2 sin(theta) cos(theta)
sin(theta) = y / r
cos(theta) = x / r
Substituting these into the polar equation, we get:
r^2 = sin(2theta)
r^2 = 2 sin(theta) cos(theta)
r^2 = 2 (y / r) (x / r)
x^2 + y^2 = 2xy
This is the rectangular form of the equation.
To convert the polar equation r = 6 / (2 - 3 sin(theta)) to rectangular form, we can first substitute sin(theta) = y / r and cos(theta) = x / r, giving:
r = 6 / (2 - 3y / r)
r(2 - 3y / r) = 6
2r - 3y = 6
This is the rectangular form of the equation.
chatgpt