For smaller jobs construction workers use a pavement roller such as the one below. To the nearest
square inch, what is the area of pavement with which the surface of
the roller will come into contact in one complete rotation?
Answers
The surface area that can be painted with one complete rotation of the roller is 1508 inches².
Surface area is defined as the total amount of area that covers the surface or outside of a three-dimensional figure.
A paint roller is in the shape of a cylinder. To determine the surface area that can be painted with one complete rotation of the roller, solve for the surface area of a cylinder without the circular bases.
SA = 2πrh
where SA = surface area
r = radius of the base = 8 inches
h = height of the cylinder = width = 30 inches
Plug in the values and solve for the surface area.
SA = 2πrh
SA = 2π(8*30)
SA = 150.7.966 = 1508 inches²
Hence, the surface area that can be painted with one complete rotation of the roller is 1508inches².
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Related Questions
Solve the system by substitution. y= 9x
y= 8x+4
Answers
the solution of a system of linear equations is (4,36) with the help of the substitution method.
The algebraic approach to solving simultaneous linear equations is known as the substitution method. The value of one variable from one equation is substituted in the second equation in this procedure, as the name implies. By doing this, a pair of linear equations are combined into a single equation with a single variable, making it simpler to solve.
The given system of equations is
[tex]y=9x\\y=8x+4[/tex]
We shall solve it with the help of the substitution method
Substitute the value of y in Equation 2
[tex]9x=8x+4\\9x-8x=4\\x=4[/tex]
Put the value of x in Equation 1
[tex]y=9*4=36[/tex]
Hence the solution of a system of linear equations is (4,36) with the help of the substitution method.
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Mega Movies hosted a film premiere on Friday night. They charged $9 for adults and $5 for children. One hundred thirty-one adults and children attended, and $1,135 was made in ticket sales. How many children and how many adults went to the film premiere?
A.
15 children; 116 adults
B.
5 children; 126 adults
C.
11 children; 120 adults
D.
8 children; 123 adults
Answers
Mega Movies hosted a film premiere on Friday night. They charged $9 for adults and $5 for children. Therefore, the equation is written as 11 children; 120 adults went to the premier.
Let children be x and adult be y
As the cost of the entry fee is in the ratio of 9 to 6
so, 9x+5y= 1135 ----------------- (1)
and x +y = 131 ------------------- (2)
Multiplying equation (2) by 9, we get:
9x + 9y = 1179
Now, subtracting equation (2) and (1), we get:
9x + 9y = 1179
9x+5y= 1135
⇒ 4y = 44
⇒ y = 11
Now putting y = 11 in equation (1) , we get:
9x+5y = 1135
⇒ 9x + 5× 11 = 1135
⇒ 9x = 1080
⇒ x = 120
Therefore, 11 children; 120 adults went to the premier.
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2+2 with square root of 9
Answers
Answer:
Therefore, 2 + 2 with a square root of 9 is equal to 8.
Step-by-step explanation:
We can simplify the expression "square root of 9" to just 3. Then, we have:
2 + 2(3) = 2 + 6 = 8
Therefore, 2 + 2 with a square root of 9 is equal to 8.
Tnx :) :) ;)
Question 1
Write an inequality and a word sentence that represent the graph. Let x
represent the unknown number.
Inequality:
Question 2
Word sentence: A number x
is
more than
1.
Answers
A word sentence that represents this inequality is "A number x is more than 1."
What is Inequality ?
In mathematics, an inequality is a statement that compares two values, expressing that one value is greater than, less than, greater than or equal to, or less than or equal to the other value. Inequalities are represented using symbols such as "<" (less than), ">" (greater than), "<=" (less than or equal to), ">
Inequality: x > 3
Word sentence: The value of x is greater than 3.
Explanation: Based on the graph, we can see that the shaded area represents all the values of x that are greater than 3. Therefore, the inequality that represents this graph is x > 3, which means that x is any number greater than 3.
Question 2:
Word sentence: A number x is more than 1.
Explanation: The inequality represented by the given graph is x > 1, which means that x is any number greater than 1.
Therefore, a word sentence that represents this inequality is "A number x is more than 1."
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Divide g by seven than multiply by six
Answers
Answer:
6/7x
Step-by-step explanation:
g/7 x 6 = 6/7 x
Answer: g / 7 x 6
Explanation:
Question 1(Multiple Choice Worth 2 points)
(Translations MC)
Triangle ABC is shown. Use the graph to answer the question.
triangle ABC on a coordinate plane with vertices at negative 8 comma 1, 0 comma 1, negative 4 comma 5
Determine the coordinates of the image if triangle ABC is translated 6 units to the right.
A′(−2, 1), B′(6, 1), C′(2, 5)
A′(−2, 1), B′(−6, 1), C′(−10, 5)
A′(−6, 7), B′(0, 7), C′(−4, 11)
A′(−6, −5), B′(0, −5), C′(−4, −1)
Answers
A′(2, 1), B′(6, 1), and C′(2, 5) are the image's positions as a result of moving the triangular 6 units to the right.
How are coordinates written down?The order of the numbers is always latitude first, semicolon after which is followed by longitude. For instance, New York City is located approximately at 40°N and 74°W in terms of latitude and longitude. The approximate coordinates for Sydney are 34°S and 150°E. When using a chart or globe, your longitude as well as latitude won't be precisely accurate.
To translate a point (x, y) to the right by 6 units, we add 6 to the x-coordinate.
So, the image of vertex A(-8, 1) would be A'(-8 + 6, 1) = A'(-2, 1).
Similarly, the image of vertex B(0, 1) would be B'(0 + 6, 1) = B'(6, 1).
And, the image of vertex C(-4, 5) would be C'(-4 + 6, 5) = C'(2, 5).
Therefore, the coordinates of the image after translating the triangle 6 units to the right are:
A′(−2, 1), B′(6, 1), C′(2, 5).
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In the equation x÷4=28 , what inverse operation would you use
Answers
Answer:
something time 4 is equal to x
Step-by-step explanation:
flip the question around
The line plots represent data collected on the travel times to school from two groups of 15 students.
A horizontal line starting at 0, with tick marks every two units up to 28. The line is labeled Minutes Traveled. There is one dot above 4,6,14, and 28. There are two dots above 10, 12, 18, and 22. There are three dots above 16. The graph is titled Bus 47 Travel Times.
A horizontal line starting at 0, with tick marks every two units up to 28. The line is labeled Minutes Traveled. There is one dot above 10,16,20, and 28. There are two dots above 8 and 14. There are three dots above18. There are four dots above 12. The graph is titled Bus 14 Travel Times.
Compare the data and use the correct measure of variability to determine which bus is the most consistent. Explain your answer.
Bus 47, with an IQR of 8
Bus 14, with an IQR of 6
Bus 47, with a range of 8
Bus 14, with a range of 6
Answers
[tex] \: [/tex]
To determine which bus is the most consistent, we need to compare the interquartile ranges (IQR) and ranges of the two data sets.
Bus 47 has an IQR of 8 and a range of 8.
Bus 14 has an IQR of 6 and a range of 6.
The IQR is a better measure of variability than the range because it is less sensitive to outliers. In this case, we can see that Bus 14 has a smaller IQR, indicating that the travel times for Bus 14 are more consistent.
Therefore, Bus 14 is the most consistent bus in terms of travel times.
Harriet's picture is 8 in x 11 in. She wants the
total area to be 108 in²2. How wide should
the border around the picture be?
Answers
Step-by-step explanation:
To find the width of the border, we first need to calculate the area of the picture.
Area of the picture = length x width = 8 in x 11 in = 88 in²
Let the width of the border be "x".
The new length of the picture will be 11+2x, and the new width will be 8+2x.
The total area of the new picture with the border will be:
total area = (11+2x) x (8+2x) = 108 in²
Expanding this equation gives:
88 + 22x + 16x + 4x² = 108
Simplifying and rearranging the terms, we get:
4x² + 38x - 20 = 0
We can solve for x using the quadratic formula:
x = (-b ± sqrt(b² - 4ac)) / 2a
where a = 4, b = 38, and c = -20.
Plugging in these values, we get:
x = (-38 ± sqrt(38² - 4(4)(-20))) / 2(4)
x = (-38 ± sqrt(1936)) / 8
x = (-38 ± 44) / 8
The two possible solutions are:
x = 1.5 or x = -5
Since a negative width for the border does not make sense, the width of the border should be 1.5 inches.
the t distribution, compared to the z distribution, is select one: a. more skewed b. more peaked for small samples but increasingly like the z distribution as n increases c. bimodal d. flatter for small sample sizes but increasingly like the z distribution as n increases
Answers
The t-distribution becomes flatter as the sample size increases.
The t distribution, compared to the z distribution, is more peaked for small samples but increasingly like the z distribution as n increases.
The t-distribution :
A t-distribution is a symmetric distribution that is similar to the standard normal distribution, also known as the z-distribution.
It is a continuous probability distribution that is used to estimate population parameters when the sample size is small (less than 30) or when the population standard deviation is unknown.
The t-distribution has more variability than the standard normal distribution due to the smaller sample size.
For a given number of degrees of freedom, the t-distribution has more density in the tails than the standard normal distribution.
The z-distribution :
The standard normal distribution is also known as the z-distribution. It is a continuous probability distribution that has a mean of zero and a standard deviation of one.
The z-distribution is bell-shaped, with a median of zero, a mode of zero, and a mean of zero.
In conclusion, the t-distribution has a peakier curve than the z-distribution, especially for small sample sizes.
However, as the sample size increases, the t-distribution approaches the z-distribution.
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all tests of hypothesis are based on the assumption that select one: a. the null hypothesis is false and should be rejected b. the observed difference is important c. the null hypothesis is true d. type i errors are more serious than type ii errors
Answers
All tests of hypothesis are based on the assumption that the null hypothesis is true. C. the null hypothesis is true is the correct option.
In hypothesis testing, two hypotheses are compared:
The null hypothesis (H0) and the alternative hypothesis (Ha).The null hypothesis is the statement that is assumed to be true, while the alternative hypothesis is the statement that the researcher is trying to prove.
For example, if the researcher wants to test whether a new drug is effective in treating a certain disease, the null hypothesis would be that the drug has no effect, while the alternative hypothesis would be that the drug is effective.
In hypothesis testing, the researcher collects data and then analyzes it using a statistical test. The test produces a p-value, which is the probability of getting the observed data if the null hypothesis is true. If the p-value is less than a pre-determined level of significance (usually 0.05), the null hypothesis is rejected and the alternative hypothesis is accepted.If the p-value is greater than the level of significance, the null hypothesis is not rejected and the alternative hypothesis is not accepted.Therefore, all tests of hypothesis are based on the assumption that the null hypothesis is true option c is correct..
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Trent created a simple mosaic in 2/5 hours.
It took Trent 3 1/2 times as long to make a second mosaic. How much of the second mosaic did Trent complete in 1 hours?
Answers
Thus, 5/7 path of the second mosaic Trent will complete in 1 hours.
Explain about the mixed fractions:A mixed number is a representation of both a whole number and a improper fraction. In most cases, it denotes a number that falls either between two whole numbers.
Divide the numerator and denominator to create a mixed number from an incorrect fraction. The solution here becomes the numerator, the remainder the denominator, and the denominator stays the same.
Given:
Time to complete simple mosaic = 2/5 hours.
Now, Time to complete second mosaic = (3 1/2)*2/5 hours.
Convert the 3 1/2 (given in mixed fraction into proper fraction).
= (3*2 + 1)/2
= 7/2
So, to complete second mosaic Trent took = (7/2)*(2/5) = 7/5 hours.
In 7/5 hours ---> 1 complete part.
In 1 hour --> 5/7 part complete
Thus, 5/7 path of the second mosaic Trent will complete in 1 hours.
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For each of the following, identify the conic section represented, then rewrite the conic section in vertex form. Submit your answers and all your work to your teacher.
[tex]x^{2} +y^{2} -2x-2y=1[/tex]
Answers
Given equation represents a circle with center at (1, 1) and vertex form [tex](x - 1)^2 + (y - 1)^2 = 2.[/tex]
What is circle?A circle is a closed shape in geometry that consists of all points that are equidistant from a central point called the center. A circle is defined by its radius, which is the distance from the center to any point on the circumference (outer boundary) of the circle. The diameter of a circle is twice the radius, and the circumference is the total distance around the circle.
According to the given information:The given equation [tex]x^2 + y^2 - 2x - 2y = 1[/tex]represents a conic section known as a circle.
To rewrite the given equation in vertex form for a circle, we need to complete the square for both x and y terms separately. The vertex form of a circle is given by:
[tex](x - h)^2 + (y - k)^2 = r^2[/tex]
where (h, k) is the center of the circle and r is the radius.
Let's complete the square for x and y terms:
[tex]x^2 - 2x + y^2 - 2y = 1[/tex]
Adding and subtracting the necessary terms to complete the square:
[tex]x^2 - 2x + 1 + y^2 - 2y + 1 = 1 + 1[/tex]
Rewriting the equation in vertex form:
[tex](x - 1)^2 + (y - 1)^2 = 2[/tex]
So, the conic section represented by the given equation is a circle with its center at (1, 1) and a radius of √2, and the vertex form of the circle is [tex](x - 1)^2 + (y - 1)^2 = 2.[/tex]
Therefore, Given equation represents a circle with center at (1, 1) and vertex form [tex](x - 1)^2 + (y - 1)^2 = 2.[/tex]
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Find the volume of the prism.
A drawing of a square prism with length, width, and height labeled start fraction 7 over 10 end fraction inch.
Answers
The volume of the prism is 0.343 cubic inches.
What is Volume ?
Volume is a measure of the amount of space occupied by a three-dimensional object. It is the quantity of space that a solid object occupies in three dimensions. Volume is often expressed in cubic units, such as cubic meters, cubic centimeters, or cubic inches , depending on the system of measurement used.
To find the volume of a rectangular prism, we multiply its length, width, and height.
In this case, the length, width, and height of the prism are all 0.7 inch. Therefore, the volume of the prism is:
Volume = (length) x (width) x (height)
= (0.7 x (0.7) x (0.7)
= 0.343 cubic inches
Therefore, the volume of the prism is 0.343 cubic inches.
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find the center and radius of the circle given by this equation: x squared space minus space 10 x space plus space y squared space plus space 6 y space minus space 30 space equals space 0 what is the center?
Answers
The center and radius of a circle given by the equation x² - 10x + y² + 6y - 30 = 0 is (5,-3) and√64 = 8 units.
When finding the center and radius of a circle given by the equation x² - 10x + y² + 6y - 30 = 0, one can use the following steps:
The first step is to rearrange the equation into the standard form, (x - a)² + (y - b)² = r². This is done by completing the square for both the x and y terms in the equation.
x² - 10x + y² + 6y - 30 = 0x² - 10x + 25 + y² + 6y + 9 - 30 = 25 + 9(x - 5)² + (y + 3)² = 64 Therefore, the center of the circle is (5,-3), and the radius is √64 = 8 units.
Explanation: Given the equation x² - 10x + y² + 6y - 30 = 0, we want to find the center and radius of the circle. The standard form of the equation of a circle with center (a,b) and radius r is (x - a)² + (y - b)² = r². We will begin by completing the square for the x terms and the y terms separately: For the x terms: x² - 10x= x² - 10x + 25 - 25= (x - 5)² - 25 For the y terms: y² + 6y= y² + 6y + 9 - 9= (y + 3)² - 9 Now we can substitute these expressions back into the original equation and simplify: x² - 10x + y² + 6y - 30 = 0(x - 5)² - 25 + (y + 3)² - 9 - 30 = 0(x - 5)² + (y + 3)² = 64 The equation is now in standard form,
which means that the center of the circle is (5,-3) and the radius is √64 = 8 units.
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Which line of music shows a reflection?
Answers
Answer:
It would be the last one.
Step-by-step explanation:
hope this is correct
Can someone plss help mee it would mean a lot
Answers
Answer:
Step-by-step explanation:
(1,-2) (-3,1)
1. y--1= -- 3/4 (x+3)
2. y= --3/4x--5/4
what's the answer for this
Answers
Answer:
y = (-1/4)x - 1
Step-by-step explanation:
We can write an equation for this line in slope-intercept form:
[tex]y=mx+b[/tex],
where [tex]m[/tex] is the slope of the line and [tex]b[/tex] is the y-coordinate of the y-intercept.
First, to solve for the slope:
slope = rise / run
slope = -1 / 4
slope = -1/4
Next, to find the y-intercept, we can see that the line crosses the y-axis (vertical axis) when y = -1.
Finally, we can form an equation using these two pieces of information.
y = (-1/4)x - 1
Find the value of x. *
43°
99°
2x°
A.28
B22.5
C.19
D.71
Answers
Answer:
A
Step-by-step explanation:
the exterior angle of a triangle is equal to the sum of the 2 opposite interior angles.
99° is an exterior angle of the triangle , then
2x + 43 = 99 ( subtract 43 from both sides )
2x = 56 ( divide both sides by 2 )
x = 28
Two of the angles in a triangle measure 88° and 84°. What is the measure of the third angle?
Answers
Answer: 8
Step-by-step explanation: a triangle has a total of 180 adding all the angles
Answer:
8°
Step-by-step explanation:
The sum of the three interior angles in a triangle is always 180°.
Third angle = 180 - (88 + 84) = 8
6(6)+18-55=118.
Help me find the answer
Answers
The following equation is given using the PEMDAS order of operations and we get the solved expression as -1.
What is BODMAS?BODMAS, also known as PEMDAS, is an acronym used to remember the order of operations in mathematics. The letters in the acronym stand for:
B - Brackets
O - Order (exponents and roots)
D - Division
M - Multiplication
A - Addition
S - Subtraction
BODMAS tells us about the order in which to perform mathematical operations in an expression. This is important because performing the operations in the wrong order can give us a different answer than what is intended.
We start by evaluating 6(6), which is 36. Next, we combine 36 and 18 to get 54.
Finally, we subtract 55 from 54 to get -1.
Therefore, the answer to the expression 6(6) + 18 - 55 is -1, not 118.
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Keilantra bought headphones online for $34. She used a coupon code to get a 40% discount. The website also applied a 5% processing fee to the price after the discount. Keilantra's headphones were less than the original price by what percent? Round to the nearest whole number.
Answers
Keilantra's headphones were less than the original price by 37%, rounded to the nearest whole number.
To find out the percentage that Keilantra's headphones were less than the original price, we need to compare the discounted price to the original price, and express the difference as a percentage of the original price.
Let's start by calculating the discounted price. Keilantra bought headphones for $34 and received a 40% discount using a coupon code:
Discounted price = Original price - Discount
Discount = 40% of Original price
Discounted price = Original price - 40% of Original price
Discounted price = 60% of Original price
Substituting the values given, we can calculate the discounted price as:
Discounted price = 0.6 x $34
Discounted price = $20.40
Now we need to apply the 5% processing fee to the discounted price:
Final price = Discounted price + Processing fee
Processing fee = 5% of Discounted price
Final price = Discounted price + 5% of Discounted price
Final price = 1.05 x Discounted price
Substituting the value we found for the discounted price, we get:
Final price = 1.05 x $20.40
Final price = $21.42
Now we can calculate the percentage that Keilantra's headphones were less than the original price:
Percentage discount = ((Original price - Final price) / Original price) x 100%
Percentage discount = (($34 - $21.42) / $34) x 100%
Percentage discount = (12.58 / 34) x 100%
Percentage discount = 0.37 x 100%
Percentage discount = 37%
Therefore, Keilantra's headphones were less than the original price by 37%, rounded to the nearest whole number.
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If JKLM is a rhombus, MK = 30, NL = 13, and m/MKL = 41°, find each measure.
Answers
The measures are: [tex]MK = 30[/tex]
[tex]NL = 13[/tex]
[tex]m/MKL = 41^\circ[/tex]
[tex]m/NLK = 41^\circ[/tex]
[tex]y \approx 17.79^\circ[/tex]
What opposite angles are congruent?In a rhombus, all sides are equal in length and opposite angles are congruent. Let's denote the measures as follows:
MK = 30 (Given)
NL = 13 (Given)
m/MKL = 41° (Given)
m/NLK = x (To be determined)
m/MKL = y (To be determined)
Since JKLM is a rhombus, we know that JM = KL and JK = ML. We can use the properties of a rhombus to find the values of x and y.
First, we can use the fact that opposite angles in a rhombus are congruent. Since [tex]m/MKL = 41^\circ[/tex] , then m/NLK (opposite angle) is also 41°.
Next, we can use the fact that diagonals of a rhombus bisect each other at a 90° angle. Since MK = 30 and NL = 13, the diagonals JL and KM intersect at their midpoint, which we can denote as O.
Using the properties of right triangles, we can form a right triangle JOM with JO = JK/2 = ML/2 (since JL is the diagonal and KM is its midpoint), OM = MK/2 = 30/2 = 15, and JM = JK = ML (since JK = ML in a rhombus).
Now we can use trigonometry to find the value of y. In right triangle JOM, we have:
[tex]sin(y) = OM/JM = 15/JM[/tex]
Since JM = KL (by the properties of a rhombus), and KL is a diagonal bisecting NL, we can use Pythagoras' theorem to find KL:
[tex]KL^2 = NL^2 + LK^2 = 13^2 + (MK/2)^2 = 13^2 + 15^2 = 169 + 225 = 394[/tex]
[tex]KL = \sqrt394[/tex]
So, we can substitute JM = KL = √394 in the equation for sin(y):
[tex]sin(y) = 15/\sqrt394[/tex]
Taking the inverse sine of both sides:
[tex]y = sin^(-1)(15/\sqrt394)[/tex]
Using a calculator, we can find the approximate value of y to be [tex]y ≈ 17.79^\circ.[/tex]
Therefore, the measures are: [tex]MK = 30[/tex]
[tex]NL = 13[/tex]
[tex]m/MKL = 41^\circ[/tex]
[tex]m/NLK = 41^\circ[/tex]
[tex]y \approx 17.79^\circ[/tex]
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Mrs. Chen bought a big tub of 250 plastic geometric pieces to use in her math classes. The pieces are all a similar size but different shapes. She randomly selects a handful of pieces from the tub. The table below shows the geometric shapes she selects.
Geometric shape Number selected
triangle 3
square 2
hexagon 2
pentagon 4
rectangle 2
Based on the data, estimate how many pentagons are in the tub.
If necessary, round your answer to the nearest whole number.
Answers
Answer:
To estimate the number of pentagons in the tub, we can use the proportion of pentagons in the sample of geometric shapes that Mrs. Chen selected and apply it to the total number of geometric shapes in the tub.
The proportion of pentagons in the sample is:
4 / (3 + 2 + 2 + 4 + 2) = 4 / 13 ≈ 0.31
We can assume that this proportion is representative of the entire tub of geometric shapes, and we can apply it to the total number of geometric shapes in the tub:
0.31 x 250 ≈ 77.5
Rounding to the nearest whole number, we can estimate that there are approximately 78 pentagons in the tub.
Therefore, the estimated number of pentagons in the tub is 78.
Any help with this please?
Answers
Answer:
223
Step-by-step explanation:
i am in 6th grade
Gerry 0.5 of the blueberries. On Monday and 25% of the blueberries on Tuesday is he 848 blueberries altogether. How many were in the box at first
Answers
There were approximately 1131 blueberries in the box at first.
To find the total number of blueberries in the box at first, follow these
First, we need to add up the percentages of blueberries Gerry ate on Monday and Tuesday. On Monday, he ate 0.5 (or 50%) of the blueberries and on Tuesday, he ate 25% of the blueberries.
So, he ate a total of 50% + 25% = 75% of the blueberries.
Since Gerry ate 848 blueberries altogether, which is 75% of the total number of blueberries in the box, we can now set up a proportion to find the total number of blueberries:
(75% of the total number of blueberries) = 848]
We can express 75% as a decimal (0.75) and set up an equation:
0.75 × (total number of blueberries) = 848
Now, to find the total number of blueberries, divide both sides of the equation by 0.75:
(total number of blueberries) = 848 / 0.75
5. Solve for the total number of blueberries:
(total number of blueberries) = 1130.67
Since there can't be a fraction of a blueberry, we can round up to the nearest whole number:
There were approximately 1131 blueberries in the box at first.
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Question
Gerry picked 0.5 of the blueberries in a box on Monday and 25% of the blueberries on Tuesday. If he picked a total of 848 blueberries altogether, how many blueberries were in the box at first?
I need help with this math problem
Answers
Answer:
This rectangle is shifted 1 unit to the left and then 4 units down. So the algebraic rule is (x - 1, y - 4).
does anyone know how to find the volume by rotating the region based on what’s in the image i uploaded?
Answers
The volume of the solid obtained by rotating the region about the y-axis is 3036π cubic units.
How to find the volume by rotating the region?
To find the volume of the solid obtained by rotating the region enclosed by the graphs of y=27-x, y=3x-9, and x=0 about the y-axis, we need to use the method of cylindrical shells.
Here the region is bounded by the graphs of y=27-x, y=3x-9, and x=0. We can see that the region is a trapezoid with vertices at (0, -9), (6, 9), (18, 9), and (27, 0).
Next, we need to determine the height and radius of each cylindrical shell. The height of each shell is the difference between the y-coordinates of the top and bottom of the shell. The radius of each shell is the x-coordinate of the shell.Let's consider a shell with x-coordinate x. The height of the shell is given by (27 - x) - (3x - 9) = 36 - 4x. The radius of the shell is x. The volume of the shell is given by 2πrhΔx, where Δx is the thickness of the shell.
Therefore, the total volume of the solid is given by:
[tex]V = \int0^6 2πx(36 - 4x)dx + \int6^18 2πx(9)dx + \int18^27 2πx(27 - x - 9)dx[/tex]
Simplifying and evaluating the integral, we get:
V = 2π(216) + 2π(243) + 2π(243/2) = 3036π
Therefore, the volume of the solid obtained by rotating the region about the y-axis is 3036π cubic units.
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Rotating the area about the y-axis yields a solid with a volume of 3036 cubic units.
The cylindrical shell method must be used to determine the solid's volume after rotating the region enclosed by the graphs of
y=27-x, y=3x-9, and x=0 about the y-axis.
In this case, the graphs of y=27-x, y=3x-9, and x=0 define the region's boundaries. The region is a trapezium, as can be seen, with vertices at (0, -9), (6, 9), (18, 9), and (27, 0).
The height and radius of each cylindrical shell must then be determined. Each shell's height is determined by the difference between its top and bottom y-coordinates.
Each shell's x-coordinate corresponds to its radius.Consider a shell with the coordinates x and y. (27 - x) - (3x - 9) = 36 - 4x gives the height of the shell. The shell has a radius of x. 2rhx, where x is the shell's thickness, gives the volume of the shell.
As a result, the solid's entire volume can be calculated using:
When we simplify and assess the integral, we obtain:
V = 2π(216) + 2π(243) + 2π(243/2)
= 3036π
As a result, the region's volume after being rotated about the y-axis is 3036 cubic units.
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An architect is designing a new windmill with four sails. In his sketch, the sails' center of rotation is the origin, (0,0), and the tip of one of the sails, point R, has coordinates (3,−6). He wants to make another sketch that shows the windmill after the sails have rotated 90° about their center of rotation. What would be the coordinates of R′?
Answers
The coordinates of R' when rotated 90° about their center of rotation is (-6, -3).
What is the geometry?Geometry is a branch of mathematics that deals with the study of shapes, sizes, relative positions of objects, and the properties of space.
The point R(3, -6) is located in the fourth quadrant as both the x-coordinate and y-coordinate are positive.
When the sails rotate 90 degrees clockwise, the point R will move to a new location R' in the third quadrant where both the x-coordinate and y-coordinate are negative.
To find the coordinates of R', we need to swap the x-coordinate and y-coordinate of point R and then negate the new x-coordinate.
That is, if R' has coordinates (x, y), then x = -6 and y = -3.
Hence, the coordinates of R' are (-6, -3).
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A line passes through the points (3,-4) and (4,-15). Write the equation in slope-intercept form using y at the output variable and x as the input variable.
Answers
The equation of the line in slope-intercept form is y = -11x + 29.
What is the formula to calculate the slope?To find the slope of given points we are using the slope-intercept formulae
The line passing through the point are (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1)
m is the slope of the line.
Putting all values in the above equation, we get:
m = (-15 - (-4)) / (4 - 3) = -11
So the slope of the line is -11
To write the equation of line the point-slope form of the equation we can use
y - y1 = m(x - x1)
Put all values into the equation
y - (-4) = -11(x - 3)
Simplifying, we get:
y + 4 = -11x + 33
Subtracting 4 from both sides, we get:
y = -11x + 29
So the equation of the line in slope-intercept form is y = -11x + 29.
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