Answer:
The areas R and S are equal.
Step-by-step explanation:
Since, the tangent line to the curve y = x³ at point A(a, a³) is given by y = 3a²(x - a) + a³. This line intersects the curve at point B(a, a³), and we can find the x-coordinate of point B by setting y = 3a²(x - a) + a³ equal to x³ to get the equation x³ - 3a²x + 3a³ - a³ = 0. Using the fact that x = a is a root of this equation, we can factor it as (x - a)³ = 0, which implies that the curve and the tangent line are tangent at point B.
The slope of the tangent line at point B is 3a², so the equation of the tangent line at point B is y = 3a²(x - a) + a³. This line intersects the curve at another point C, and we can find the x-coordinate of point C by setting y = 3a²(x - a) + a³ equal to x³ to get the equation x³ - 3a²x + 3a³ - a³ = 0. Using the fact that x = a is a root of this equation, we can factor it as (x - a)³ = 0, which implies that the curve and the tangent line are tangent at point C.
The region bounded by the curve and the tangent line at point A is a triangle with base a - (a - a) = 0 and height a³ - a³ = 0, so its area is 0. The region bounded by the curve and the tangent line at point B is a triangle with base a - a = 0 and height a³ - a³ = 0, so its area is also 0. Therefore, the areas R and S are both equal to 0, which means they are equal to each other.
The graph shows the distribution of the amount of chicken (in ounces) that adults eat in one sitting. The distribution is approximately Normal, with a mean of 8 ounces and a standard deviation of 1.2 ounces.
A graph titled Chicken Consumption has amount (ounces) on the x-axis, going from 3.2 to 12.8 in increments of 1.2. The highest point of the curve is at 8.
What percentage of adults eat more than 10.4 ounces of chicken in one sitting?
A. 2.5%
B. 47.5%
C. 95%
D. 97.5%
The percentage of adults who eat more than 10.4 ounces of chicken in one sitting is 2.5%. The correct option is A.
We know that the distribution is approximately normal with a mean of 8 ounces and a standard deviation of 1.2 ounces. We want to find the percentage of adults that eat more than 10.4 ounces of chicken in one sitting.
To solve this problem, we can use the standard normal distribution table or a calculator with the normal distribution function.
Using the calculator, we can standardize the value 10.4 using the formula:
[tex]z = \dfrac{(x - \mu)} { \sigma}[/tex]
where x is the value we want to standardize, μ is the mean, and σ is the standard deviation.
In this case, we have:
[tex]z = \dfrac{(10.4 - 8) }{1.2} = 2[/tex]
We can then use the calculator to find the percentage of the distribution that lies above the z-score of 2. This is approximately 2.5%.
Therefore, the answer is A. 2.5%.
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Danielle is saving to buy a bedroom set that cost $2,976. If she saves for 24 weeks, how much will she need to save each week?
Answer:
$124 per week
Step-by-step explanation:
To fins how much she needs to save each week, divide the total cost by the amount of time. In this problem, divide $2,976 by 24 weeks to find the amount she needs to save.
$2976/24=$124 per week
Apiece of Wire 80cm long is cut into the two parts and bent each part to Form a square the tota area of the two squaresquare is 250 Find perimiter of each Square side length of each Square
The perimeter of the square with side of 5 cm is 20 cm.
The perimeter of the square with side length of 15 cm is 60 cm.
What is the area of each square?The area of each square is calculated as follows;
Let the dimension of first square = x
Let the dimension of second square = y
x² + y² = 250 ---- (1)
4(x + y) = 80 ----- (2)
x + y = 80/4
x + y = 20
x = 20 - y
Substitute x in equation (1)
(20 - y)² + y² = 250
400 - 40y + y² + y² = 250
2y² - 40y + 150 = 0
y² - 20y + 75 = 0
solve for y, using formula method;
y = 15 or 5
Solve for x;
x = 20 -15 = 5 or
x = 20 - 5 = 15
The perimeter of first = 4x = 4 x 5 = 20 cm
The perimeter of the second = 4y = 4 x 15 = 60 cm
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What is 3/11 simplified
pls help me and explain as well because I rlly don’t understand this
Step-by-step explanation:
A linear function is a function where we have a constant rate of change.
The equation of a linear function is
[tex]y = mx + b[/tex]
where m is the slope (change in y/change in x)
and b is the y intercept.
Note: Sometimes our variables will different names, but
Slope is just change in dependent variable/change in independent variable.
A linear function has a constant slope throughout.
In this problem,
t is the independent variable
W is the dependent variable
So our point will. be
(t,W)
where t is number of weeks after planted
and W is weight
So the slope would be
Change in W/Change in t.
We know at 3 weeks, the pumpkin weighed 141
So this point is
(3,141)
We know 7 weeks, later, the pumpkin weighed 372 so
(10,372)
A. The weight the pumpkin gain each week is our slope so
[tex] \frac{372 - 141}{7} [/tex]
[tex] \frac{231}{7} = 33[/tex]
So the pumpkin gained 33 pounds each week.
Part 2: Is in slope intercept form,
and we know the slope,33,
[tex]w = 33t + b[/tex]
We don't know what b is.
Let plug in a coordinate pair, and solve for b.
Let's use (3,141)
[tex]141 = 33(3) + b[/tex]
[tex]141 = 99 + b[/tex]
[tex]42 = b[/tex]
So our equation is
[tex]w = 33t + 42[/tex]
Hope this helps
The weight that the pumpkin gained per week can be found to be 33 pounds.
The equation to show the relationship between pumpkin weight and elapsed time is W = 33 t + 42
How to find the weight gained ?The weight gained by the pumpkin per week can be found by the formula :
= (Weight gained after 7 more weeks - Weight after 3 weeks ) / 7 weeks
= ( 372 - 141 ) / 4
= 33 pounds
This means the original weight of the pumpkin was:
= 141 - ( 33 x 3 weeks )
= 42 pounds
The equation to show the relationship between pumpkin growth and weeks is therefore:
W = Growth per week x Number of weeks + original weight
W = 33 x + 42
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Mark and his three friends ate out at Applebee's. Their bill totaled $52.35. If they left the server a 20% tip, how much would each person pay splitting the bill evenly?
Step-by-step explanation:
so the total with the tip is $62.82 then divide by 3 so I believe your answer is $20.94
$15.70
Step-by-step explanation:
20% tip on $52.35 is $10.47
$10.47 + $52.35 = $62.82
$62.82 ÷ 4(him and 3 friends) = $15.70
So, each person would pay $15.70 if they split the bill evenly.
_
5
Given f(x)=x+7, if f(x) = -13, find x.
2
Helpp
Answer: -20
Step-by-step explanation: I got negative twenty because if f(x) equals negative thirteen the we add negative seven because we need to flip the sign since it is negative and when you add you get negative twenty
To check, simply solve the eqation the way it first was, for example, f(x)=-20+7 which would be correct, equaling -13
Hopefully this helped, I' sorry if i am wrong
Given the function g(x) = -x² − 9x + 27, determine the average rate of change
of the function over the interval -9 ≤x≤-1.
The rate of change of the function g(x) in the interval -9 ≤ x ≤ -1 is 1
How to determine the rate of change?The rate of change of a function y = f(x) on the interval [a, b] is given by the formula below:
R = (f(b) - f(a))/(b - a)
Here the function is g(x) = -x² − 9x + 27 and the interval is [-9, -1], we can evaluate g(x) in both ends of our interval, then we will get the two values needed for the numerator, here we will get:
g(-1) = -(-1)² − 9*-1 + 27 = -1 + 9 + 27 = 35
g(-9) = -(-9)² -9*-9 + 27 = 27
Replacing that in the formula we can get the rate of change is:
R = (35 - 27)/(-1 + 9) = 8/8 = 1
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Jared's class took a survey to see how many students owned a trampoline. Of the 24 students in the class, 14 students said they owned a trampoline. If you chose a student at random from Jared's class, what is the probability that the student does not own a trampoline?
The probability that a student chosen at random from Jared's class does not own a trampoline is 5/12.
The probability that the student does not own a trampoline is equal to the number of students who do not own a trampoline divided by the total number of students in the class.
The number of students who do not own a trampoline is equal to the total number of students in the class minus the number of students who own a trampoline:
24 - 14 = 10
Therefore, the probability that a student chosen at random from Jared's class does not own a trampoline is:
P(not owning a trampoline) = 10/24
Simplifying the fraction,
P(not owning a trampoline) = 5/12
Therefore, the probability that a student chosen at random from Jared's class does not own a trampoline is 5/12.
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On a recent math test, the mean score was 75 and the standard deviation 5. What is the 25th percentile for the math scores
The 25th percentile for the math scores is 71.65.
How to determine the 25th percentile for the math scores?In Mathematics and Geometry, the z-score of a given sample size or data set can be calculated by using this formula:
Z-score, z = (x - μ)/σ
Where:
σ represents the standard deviation.x represents the sample score.μ represents the mean score.Note: 25th percentile for the math scores is the same as the first quartile for the math scores.
Based on the z-score table A, 25th percentile (0.25) is equal to a z-score of -0.67;
0.2514 = -0.67
By substituting the parameters, we have:
Z-score, z = (x - μ)/σ
-0.67 = (x - 75)/5
x = 75 - 3.35
x = 71.65
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graph the following linear inequalities; <-2x+7
The graph of the linear inequality can be seen in the image at the end.
How to graph the linear inequality?I can assume that the linear inequality you want to graph is:
y < -2x + 7
To graph this inequality, what we need to do is graph the line:
y = -2x + 7
With a dashed line, and then shade all the region below that line.
To graph the line just find two points in the line, for example:
if x = 0
y = -2*0 + 7 = 7 ----> (0, 7)
if x = 1
y = -2*1 + 7 = (1, 5)
now graph the points and connect them with a dashed line, and then shade all the region below.
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If you were to graph 9x + 5y > 1 , where would you shade according to the slope of the line? The shading would occur above the solid line of the slope The shading would occur below the solid line of the slope. The shading would occur below the dashed line of the slope. The shading would occur above the dashed line of the slope.
The requried shading would occur above the solid line of the slope.
To graph the inequality 9x + 5y > 1, we first need to graph the equation 9x + 5y = 1, which is the equation of the line that separates the two regions determined by the inequality.
To graph this line, we can solve for y in terms of x:
9x + 5y = 1
y = (-9/5)x + 1/5
This is the equation of a line with slope -9/5 and y-intercept 1/5. We can plot this line on a coordinated plane. Since the slope of the line is negative, the shading should occur above the line, which means the correct answer is: The shading would occur above the solid line of the slope.
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Johan wants to make some small cakes.
He finds a recipe that says he needs 360 grams of flour to make 15 small cakes.
Johan has 0.85 kg of flour.
Johan works out how much flour he would need to make 38 small cakes, using the
information given in the recipe.
Does Johan have enough flour, according to the recipe, to make 38 small cakes?
Show your working clearly.
The amount of flour that will be needed for 38 cakes is 912 grams therefore Johan does not have enough amount of flour
What is word problem?A word problem is a mathematical exercise where significant background information on the problem is presented in ordinary language rather than in mathematical notation.
360 grams is needed for 15 small cakes
therefore for 1 small cake, 369/15 = 24grams will be needed
therefore for 38 small cakes = 38× 24 = 912 grams of flour will be needed.
Johan has 0.85 kg of flours which is equivalent to 0.85 × 1000grams = 850 grams of flour.
Since 912 grams will be needed for 38 small cakes , it shows that Johan does not have enough amount of flour for 38 small cakes.
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What is the probability the he will choose a red marble
The probability that Tom will choose a red marble from the bag is 1/9.
Given that,
Tom is choosing at random from a bag of colored marbles.
Odds in favor = number of success : number of failures
Odds of getting a red marble = 1 : 8
This means that there ratio of red marbles to other marbles = 1 : 8
Number of red marbles = 1
Total number of marbles = 1 + 8 = 9
Probability of choosing a red marble = 1/9
Hence the required probability is 1/9.
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Can someone help me please
Applying the Solution to a 3X3 System
At a family reunion, there only blood relatives, consisting of children, parents, and grandparents, in attendance. There were 400 people total. There were twice as many parents as grandparents, and 50 more children than parents. How many children, parents, and Grandparents were in attendance?
Show up all steps please
There are 70 grandparents, 140 parents, and 190 children in attendance at the family reunion
Given data ,
Let's denote the number of grandparents as "g", the number of parents as "p", and the number of children as "c".
Given that there were 400 people total at the family reunion, we can write the equation:
g + p + c = 400
Now , there were twice as many parents as grandparents, so we can write:
p = 2g
And we are given that there were 50 more children than parents, so we can write:
c = p + 50
Now we can use the equations (1), (2), and (3) to solve for the values of g, p, and c.
On simplifying the equation , we get
g + 2g + (2g + 50) = 400
5g + 50 = 400
5g = 400 - 50
5g = 350
g = 350 / 5
g = 70
Now we can substitute the value of g into (2) to find the value of p:
p = 2g = 2 x 70 = 140
And we can substitute the value of p into (3) to find the value of c:
c = p + 50 = 140 + 50 = 190
Hence , there were 70 grandparents, 140 parents, and 190 children in attendance at the family reunion
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HELP ASSAP 20 POinTSSSSSSSSSSSSSS
Step-by-step explanation:
use the formula V=1/3hπr².
V=1/3×3 π 2×2
V=1/3×12π
V=4πcm3
When graphing equations on paper we must show the x andy axis, numbers for these axis and the points for the
graph
True
False
Pls help, this is for geometry
I'm not 100 percent sure so please only rate after you see if it's right.
ANSWER: $1090
STEP 1:
The volume of one brick is given by multiplying its length, width, and height:
V = (7 1/4) in x 3 in x (2 1/4) in
V = (29/4) in x 3 in x (9/4) in
V = 2187/16 cubic inches
STEP 2:
The cost of one brick is given by multiplying the volume by the cost per cubic inch:
C = (2187/16 cubic inches) x $0.05/in^3
C = $1.09/brick
STEP 3: To find the cost of 1000 bricks, we simply multiply the cost of one brick by 1000:
Cost of 1000 bricks = 1000 x $1.09/brick
Cost of 1000 bricks = $1090
Therefore, the cost of 1000 bricks is $1090, rounded to the nearest cent.
Segment AB is perpendicular to the circle centered at O. The length of AB is 2.5 units and the length of CB is 1.5 units. Find the exact length of the radius of this circle.
The exact length of the radius of this circle is 2.69 units.
Given that, the length of AB is 2.5 units and the length of CB is 1.5 units.
Since AB is perpendicular to the circle, the triangle ABC is a right triangle. By the Pythagorean theorem, the radius of the circle can be found by solving:
r² = (2.5)² + (1.5)²
r² = 7.25
r = √7.25
r = 2.69 units
Therefore, the exact length of the radius of this circle is 2.69 units.
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Think of a number puzzle
The equation is: (2x + 8) / 4 - 3 = x - 7
The solution is x = 12
How to solve the puzzleLet x be the number you thought of at the beginning. Following the steps:
Multiply by 2: 2x
Add 8: 2x + 8
Divide by 4: (2x + 8) / 4
Subtract 3: (2x + 8) / 4 - 3
The final answer is seven less than the number you thought of at the beginning: (2x + 8) / 4 - 3 = x - 7
The equation is: (2x + 8) / 4 - 3 = x - 7
2. (2x + 8) / 4 - 3 = x - 7
(1/2)x + 2 - 3 = x - 7
(1/2)x - 1 = x - 7
Now add 1 to both sides:
(1/2)x = x - 6
Subtract (1/2)x from both sides:
-1/2x = -6
Now multiply both sides by -2:
x = 12
3. Properties used to solve the puzzle
Addition Property of Equality
Multiplication Property of Equality
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PLEASE HELP ME :( PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME PLEASE HELP ME
Answer:
Step-by-step explanation:
...Negative.
Answer:
Negative
Step-by-step explanation:
What is the volume of a hemisphere with a radius of 9.8 in, rounded to the nearest tenth of a cubic inch?
The volume of the hemisphere is approximately 1970.2 cubic inches.
Given that:
Radius, r = 9.8 inches
The formula for the volume of a hemisphere is given as:
V = (2/3) x π x r³
V = (2/3) x π x (9.8 in)³
V = (2/3) x 3.14 x 941.192
V = 1970.2 cubic inches
Therefore, the volume of the hemisphere is approximately 1970.2 cubic inches.
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What is 0.007 divided by 0.498 to nearest tenth
Answer:
0.0
Step-by-step explanation:
To find 0.007 divided by 0.498, we can perform the following calculation:
0.007 / 0.498 ≈ 0.0141
Rounding this to the nearest tenth gives:
0.0141 ≈ 0.0
Therefore, the result, rounded to the nearest tenth, is 0.0.
Which statement could be made based on the diagram below?
A) m∠3 + m∠6 = 90
B) ∠3 = ∠6
C) ∠3 = ∠5
D) m∠4 + m∠5 = 180
Answer:
A) angle 3 plus angle 6 = 90°
Step-by-step explanation:
just think about it : to the left of m at the intersection with n is 90°. so, to the right it must be 90° too.
because one side of a line represents always 180° for all angles around a single point.
A right triangle has side lengths 6 centimeters and 8 centimeters. Is it possible for there to be more than one right triangle with whole-number side lengths that includes the given side lengths? Explain.
The given side lengths, 6 cm and 8 cm, satisfy the Pythagorean theorem because $6^2 + 8^2 = 100 = 10^2$, which means that the triangle is a right triangle.
To determine if there is more than one right triangle with whole-number side lengths that includes these given side lengths, we need to check if there exist other pairs of whole numbers that satisfy the Pythagorean theorem and include 6 cm and 8 cm as two of their sides.
Let's call the third side of the right triangle x. Then we have:
62+82=x26^2 + 8^2 = x^262+82=x2
which simplifies to:
36+64=x236 + 64 = x^236+64=x2
100=x2100 = x^2100=x2
x=100=10x = \sqrt{100} = 10x=100
=10
So the only other possible triangle with whole-number side lengths that includes the given side lengths has side lengths of 6 cm, 8 cm, and 10 cm.
Therefore, there is only one other right triangle with whole-number side lengths that includes the given side lengths, and it is unique.
i-Ready
Dilations and Similarity-Quiz-Level H
Rays are shown from point D, the center of dilation, through the vertices of AABC.
Dilate ABC by a scale factor of 3.
A
1
C
D
B
1
Based on the information provided dilate triangle ABC by a scale factor of 3, with point D as the center of dilation.
To perform a dilation, each point in the original figure is transformed by multiplying its coordinates by the scale factor and then translating the figure according to the center of dilation. In this case, each point in triangle ABC will be multiplied by a scale factor of 3, with point D as the center of dilation.
To dilate the triangle by a scale factor of 3, you can follow these steps
Draw rays from point D to the vertices of triangle ABC.Measure the distance between each vertex and point D. This distance will be the radius of the dilation.Multiply the x-coordinate and y-coordinate of each vertex by 3, since the scale factor is 3.Use the measured distances as a guide to draw the dilated triangle, with each vertex located at the end of a ray from point D.Hence, if the scale factor is greater than 1, the dilated figure will be larger than the original. If the scale factor is between 0 and 1, the dilated figure will be smaller than the original. If the scale factor is negative, the figure will be reflected across the center of dilation.
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5(10v-8w+4) as a distributive property
Using distributive property, the solution to the given expression is: 50v - 40w + 20
How to use distributive property?The Distributive Property is a term which is an algebra property which is used to multiply a single term and two or more terms inside a set of parentheses.
For example:
4(3 + 5)
According to the distributive property, you first have to multiply out the bracket to get:
4*3 + 4*5 = 12 + 20
= 32
Thus:
5(10v - 8w + 4) = (5 * 10v) - (5 * 8w) + (5 * 4)
= 50v - 40w + 20
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The population of a particular country was 23 million in 1981; in 1987, it was 32 million. The
exponential growth function A =23ekt describes the population of this country t years after 1981. Use
the fact that 6 years after 1981 the population increased by 9 million to find k to three decimal places.
K's value is around 0.055 to three decimal places.
Let's first use the given information to set up two equations:
A(0) = 23 (the initial population in millions)
A(6) = 32 (the population 6 years later in millions)
Substituting these values into the exponential growth function A = 23e^(kt), we get:
[tex]23 = 23e^(k0) = > e^{(k0)} = 1\\32 = 23e^{k*6}[/tex]
Dividing the second equation by the first, we get:
[tex]32/23 = e^{6k}[/tex]
Taking the natural logarithm of both sides, we get:
ln(32/23) = 6k
Solving for k, we get:
k = ln(32/23)/6
k ≈ 0.055
Rounding to three decimal places, we get k ≈ 0.055.
Therefore, the value of k to three decimal places is approximately 0.055.
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Amelie travelled to her friend's house and her journey is shown on the distance-time graph below. She travelled the final 4 hours of her journey at a constant speed of 60 km/h. Work out the value of d. Distance travelled (km) 270- 0 3 Time (hours) -
The total distance traveled by Amelia to her friend's house is d = 510 miles
Given data ,
Let the distance traveled by Amelia to her friend's house be d
Now , the speed of the final 4 hours of her journey = 60 km/hr
So , the distance traveled in the final 4 hours = 4 x 60 = 240 km
And , the initial distance traveled = 270 km
So , from the graph
The total distance d = 270 + 240
d = 510 miles
Hence , the distance traveled by Amelia = 510 miles
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arch
A group of students is given a 10 by 10 grid to cut into
individual unit squares. The challenge is to create two
squares using all of the unit squares. Their teacher
states that after the two new squares are formed, one
should have a side length two units greater than the
other.
Which equation represents x, the side length of the
greater square?
O x² + (x-2)2 = 10
O x² + 2x² = 10
O x² + (x-2)2 = 100
Ox²+2x² = 100
The equation that represents x, the side length of the greater square is this: x² + (x-2)²= 100.
Which equation represents x?The equation that represents x, the greater side length is x² + (x-2)²= 100. According to the instruction from the teacher, one of the squares should have a side that is two units greater than the first side.
So, if the first side was x, then the new side will be x2 and the same will be true of side, x -2 which now becomes (x-2)². Of course, multiplying a side 10 by 10 will also give 100. So, the correct equation will be C.
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