The area of a rectangle that is inscribed with its base on the x-axis and its upper corners on the parabola [tex]y=c-x^2[/tex] is [tex]4(\frac{c}{3})^\frac{3}{2}[/tex].
A rectangle has a width [tex]2x[/tex] ⇒ w= [tex]2x[/tex]
height of the rectangle is [tex]c-x^2[/tex] ⇒ [tex]y=c-x^2[/tex]
Area=width multiplied by the height
⇒[tex]area=width*height[/tex]
we maximize Area by setting [tex]\frac{dA}{dx} =0[/tex] .
[tex]\frac{d(2c-2x^3)}{dx} =0\\\\2c-6x^2=0\\\\x^2=\frac{1}{3}c\\ \\x=\sqrt{\frac{c}{3} }[/tex]
substituting back the value of x in the formula of the area we get
[tex]area=2\sqrt{\frac{c}{3} } *2\frac{c}{3} \\\\area=4(\frac{c}{3})^\frac{3}{2}[/tex]
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find two numbers whose difference is 86 and whose product is a minimum. smaller number larger number
The two numbers whose difference is 86 and whose product is a minimum. The numbers are (43,-43) and therefore, the smaller number is -43 and largest number is 43.
Given :
Two numbers whose difference is 86 and whose product is a minimum.
To find : The numbers
Let the largest number be x,
and smallest number be y.
According to question,
Two numbers whose difference is 86.
i.e. x - y = 86 -----------------------------(1)
Product of two number is minimum.
i.e. xy = 86 -------------------------------- (2)
where P is minimum.
Now substitute x from (1) in (2),
x (x -86) = P
⇒ y(x) = x² - 86x = P ------------------------------ (3)
Differentiating Equation (3), we get:
y'(x) = 2x - 86
and y'(x) = 0
After subtracting we get:
2x - 86 = 0
⇒ x = 86 /2
⇒ x = 43
The vertex of the equation occur at x = 43.
Putting the value of x = 43 in equation (1) , we get:
48 - y = 86
⇒ -y = 86 - 48
⇒ y = -43
So, the value of y is - 43.
Therefore, the numbers are (43,-43).
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Write the equation of the graph sketch below
The equation of the graph is f(x)=-2x+5.
What is Slope of Line?The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.
The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.
The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is
m=y₂-y₁/x₂-x₁
Let us take any two points from the graph to find the equation of the graph.
(-1, 3) and (2, -3)
Let us find the slope of line.
m=-3-3/2+1
m=-6/3=-2
Slope of line is -2 and now we find the y intercept.
y=-2x+b
3=-2(-1)+b
3+2=b
5=b
Hence, f(x)=-2x+5 is the equation of the graph.
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brainly question 5: charlie has $125 when he goes to the school carnival. he spends $15 to enter the carnival and $40 on food. games at the carnival cost $5 each to play. which function can be used to determine how much money he has left over after x rides?
The function that can be used to determine the amount of money he has left over after x rides is y = 70 - 5x .
In the question ,
it is given that ,
the amount that Charlie has to spend on the school carnival is = $125 ,
the amount spend on entry is = $15 ,
the amount spend on food is = $40 ,
the cost for each ride is = $5 ,
so , let the number of rides be = "x"
So , let y represents the amount of money left with Charlie .
According to the question ,
the function that represents the money left is
y = 125 - (15 + 40 + 5x)
y = 125 - (55 + 5x)
y = 125 - 55 - 5x
y = 70 - 5x
Therefore , the function is y = 70 - 5x .
The given question is incomplete , the complete question is
Charlie has $125 when he goes to the school carnival. he spends $15 to enter the carnival and $40 on food. rides at the carnival cost $5 each to play. which function can be used to determine how much money he has left over after x rides ?
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HJ is extended through point J to point K, m∠JHI =(x+12) m∠HIJ=(x−3) and
m∠IJK =(5x−6) Find m∠IJK.
The value of m∠IJK is 19.
What is a triangle?It is a two-dimensional figure which has three sides and the sum of the three angles is equal to 180 degrees.
We have,
I
/ \
/ \
/ \
H / \ J K
Now,
m∠JHI = (x+12)
m∠HIJ = (x−3)
m∠IJK = (5x−6)
The sum of the angles in the triangle IHJ is 180 degrees.
m∠JHI + m∠HIJ + m∠IJH = 180
x + 12 + x - 3 + m∠IJH = 180
2x + 9 + m∠IJH = 180
2x + m∠IJH = 180 - 9
2x + m∠IJH = 171
m∠IJH = 171 - 2x ____(1)
Now,
m∠IJK = 180 - m∠IJH
5x - 6 = 180 - m∠IJH
m∠IJH = 180 - 5x + 6
m∠IJH = 186 - 5x _____(2)
Now,
From (1 ) and (2) we get,
186 - 5x = 171 - 2x
186 - 171 = 5x - 2x
15 = 3x
x = 5
Now,
The value of m∠IJK.
= 5x - 6
= 5 x 5 - 6
= 25 - 6
= 19
Thus,
m∠IJK value is 19.
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Which equation represents a line passing through the point (8, 16) with a slope of 3?
A y=3x – 15
B)
y=3x + 33
C)
y=3x – 8
D)
y=3x + 9
Answer:
C) [tex]y=3x-8[/tex]
Step-by-step explanation:
[tex]y-16=3(x-8) \\ \\ y-16=3x-24 \\ \\ y=3x-8[/tex]
Katherine has 9½ cups of yogurt to make smoothies. Each smoothie uses 1/2
cup of yogurt. How many smoothies can Katherine make with the yogurt?
Answer:
19
Step-by-step explanation:
9 1/2 divided by 1/2 is 19
For what slit-width-to-wavelength ratio does the first minimum of a single-slit diffraction pattern appear at (a) 30 degrees (b) 60 degrees, and (c) 90 degrees? Hint: The small-angle approximation is not valid.
The slit-width-to-wavelength ratio for 30 degrees will be 2; for 60 degrees will be 0.87, and lastly, for 90 degrees angles will be 1.
The computation of slit-width-to-wavelength ratio for each of the angles after assuming that the small-angle approximation as invalid, has been given, as below,
sin θ = λ / a
sin30∘ = λ / a
0.5 = λ / a
a / λ = 2
Similarly,
sin θ = λ / a
sin60∘ = λ / a
0.87 = λ / a
a / λ = 0.87
And lastly, for 90 degrees
sin θ = λ / a
sin90∘ = λ / a
1 = λ / a
a / λ = 1
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What is the limit X?
The limit of x/|x| as x approaches 0 does not exists.
We need to find the limit of x/|x| as x approaches 0.
Consider left hand limit.
lim x -> 0- (x/|x|)
x/|x| = -1 for all negative values of x.
So, lim x -> 0- (x/|x|) = -1
Consider right hand limit.
lim x -> 0+ (x/|x|)
x/|x| = 1 for all positive values of x.
So, lim x -> 0+ (x/|x|) = 1
lim x -> 0- (x/|x|) ≠ lim x -> 0+ (x/|x|)
Therefore, the limit of x/|x| does not exists.
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The complete question is:
What is the limit of x/|x| as x approaches 0?
which figure has rotational symmetry and line symmetry?
Answer:
Figure 1 (The Star)
Step-by-step explanation:
Line symmetry is symmetry created by a line that divides a shape. Only the "star" (Figure 1) and the "A" (Figure 2) can be split through the center into two identical halves, therefore we can rule out Figures 3 & 4. Rotational symmetry is when a shape looks the same after a partial turn. Rotational symmetry is only true for the "star" (Figure 1) which will be symmetric every 72-degree (360/5) turn. However, the "A" (Figure 2) can only be flipped 180 degrees to remain symmetric.
Therefore Figure 1 is the only figure with both rotational and line symmetry.
which inequality represents the graph?
The inequality graphed on the given image is the one in option B, the inequality is:
x > -2.
Which inequality represents the graph?On the graph, we can see an inequality. We can see a vertical line and all the region to the right of that line is shaded.
The vertical line is x = -2
If the right side of that is shaded, then all the points (x, y) with a value of x larger than -2 are solutions, then the inequality is just:
x > -2
Then the correct option is B.
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What does it mean to dilate a rectangle?
To dilate a rectangle means to either expand or contract the size of rectangle without changing it's shape .
Dilation is where the polygon grows or shrinks but keeps the same overall shape.
To dilate a rectangle means to either expanding or contracting the rectangle without changing it's shape .
For example: Assume that, the origin (0, 0) be the center of dilation in the coordinate plane.
Let ABCD be a Rectangle in the coordinate plane. The points in the coordinate planes are A(-2, 2), B(3, 2), C(3, -2), D(-2, -2)
To dilate this rectangle by factor 3 , then every coordinate point of the original triangle will multiplied by the scale factor
Coordinates of new rectangles : A'(-4,4) , B'( 6, 4) ,C(6, -4) and D'(-4, -4)
Therefore, When we dilate a rectangle by scale factor 2 ,
The rectangle has enlarge it's size .
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PLEASE ANSWER ASAP! WILL GIVE 35 POINTS!
a. AB = FG, AC = FH, BC = GH
b. Remember that FG is the same as AB. This means that their values are equivalent. Therefore, FG = 9.
c. HG is the same as CB. If CB is 12, then HG must also be 12.
d. To solve for x, we'll need to set up an equation. We can do this by taking one of our known pairs of congruent sides and setting them equal to each other.
AB = FG
AB = 9
FG = 2x + 1
9 = 2x + 1
2x = 8
x = 4
Hope this helps!
lesson practice what is the approximate volume of this cylinder? what is the closest to the volume of a bowling ball with a diameter of 21.8 centimeters?
volume of ball = 5421.85[tex]cm^{3}[/tex]
Volume of a Sphere
A sphere is a set of points in space that are a given distance r from the center.
The volume of a 3 -dimensional solid is the amount of space it occupies. Volume is measured in cubic units( in3,ft3,cm3,m3 , et cetera). Be sure that all of the measurements are in the same unit before computing the volume.
The volume V of a sphere is four-thirds times pi times the radius cubed.
V=4/3πr^3
The volume of a hemisphere is one-half the volume of the related sphere.
Given that
diameter of ball = 21.8 cm
now we need to find the volume of ball
volume of the sphere =[tex]\frac{4}{3}[/tex] [tex]\pi[/tex][tex]r^{3}[/tex]
diameter = 21.8cm
radius = diameter/2
= 21.8/2
=10.9
now substitute the value of radius in the volume formula
volume of ball =[tex]\frac{4}{3}[/tex][tex]\pi[/tex][tex](10.9)^{3}[/tex]
volume of ball = 5421.85[tex]cm^{3}[/tex]
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Jayden practices the piano 420 minutes in 2 weeks. Assuming he practices the same amount every week, how many minutes would he practice in 3 weeks?
Answer: Jayden would practice for 630 minutes.
Step-by-step explanation:
You know that he practices for 420 minutes in 2 weeks. Divide them by each other. [tex]\sqrt[2]{420}=210[/tex]
If one week is 210 minutes, 210x3=630 minutes.
At 5pm, you count 26,300 alien bacteria in your petri dish. If the growth rate is 2. 7%, how many bacteria will there be at midnight?.
The count increase in the alien bacteria at the midnight is 31692.
Here we have to find the count of alien bacteria at midnight.
Data given:
Count of alien bacteria at 5 pm = 26,300
Percentage increase in bacteria per hour = 2.7%
increase = 1 + 2.7/100
= 102.7
time = 12 - 5 = 7
The count of alien bacteria at midnight = 26,300 ×( 1.207 [tex])^{7}[/tex]
= 31692
Therefore the alien bacteria at midnight is 31692.
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Write the equation of a line in slope intercept form that has a slope of 1/2
and passes through the point (-2,3).
Answer:
[tex]y=\frac{1}{2}x+4[/tex]
Step-by-step explanation:
[tex]y-3=\frac{1}{2}(x+2) \\ \\ y-3=\frac{1}{2}x+1 \\ \\ y=\frac{1}{2}x+4[/tex]
Determine if the following relations are functions or non functions
Please help me!!!!
keeping in mind that a function does not have any X-Repeats, that is, the 1st coordinate in every pair never repeats, that is, the INPUT never repeats, Check the picture below.
now, to check on the graph if it's a function or not, simply recall the vertical line test, if you can draw a vertical line on the graph and it only touches it ONCE, then it's a function, if however a vertical line touches it more than once, then is not.
A thermometer shows the temperature in degrees Celsius (°C) and degrees Fahrenheit (°F). One day, it shows 68°F and 20°C. The next day, it shows 50°F and 10°C.
Determine if there is a proportional relationship between the temperature in degrees Celsius and the temperature in degrees Fahrenheit.
Answer:
To determine if there is a proportional relationship between the temperature in degrees Celsius and the temperature in degrees Fahrenheit, we need to see if the ratio of the temperature in degrees Celsius to the temperature in degrees Fahrenheit is constant.
On the first day, the ratio of the temperature in degrees Celsius to the temperature in degrees Fahrenheit is 20°C / 68°F = 5/17.
On the second day, the ratio of the temperature in degrees Celsius to the temperature in degrees Fahrenheit is 10°C / 50°F = 1/5.
Since the ratio of the temperature in degrees Celsius to the temperature in degrees Fahrenheit is different on the two days, there is not a proportional relationship between the temperature in degrees Celsius and the temperature in degrees Fahrenheit.
Step-by-step explanation:
Since both data points satisfy the proportional relationship equation, we can conclude that there is a proportional relationship between the temperature in degrees Celsius and the temperature in degrees Fahrenheit.
Do we have a proportional relationship between the temperature in degrees Celsius and the temperature in degrees Fahrenheit?To determine if there is a proportional relationship between the temperature in degrees Celsius (°C) and degrees Fahrenheit (°F), we can check if the data points given for both temperatures satisfy the proportional relationship equation:
°F = (9/5) * °C + 32
Let's calculate the proportional relationship using the given data points:
1. First data point: 68°F and 20°C
°F = (9/5) * 20 + 32
°F = 36 + 32
°F = 68
The first data point (68°F and 20°C) satisfies the proportional relationship equation.
2. Second data point: 50°F and 10°C
°F = (9/5) * 10 + 32
°F = 18 + 32
°F = 50
The second data point (50°F and 10°C) also satisfies the proportional relationship equation.
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I need help asap!!! What is the answer to this question:
Answer:
15/4
Step-by-step explanation:
[tex]\ell=5/2 \\ \\ w=1 \\ \\ h=3/2 \\ \\ V=lwh=(5/2)(1)(3/2)=15/4[/tex]
Answer:
3.75 so i believe the answer might be 15/4?
Step-by-step explanation:
im sorry if this is not the right answer
The selling price of an item is $360. It is marked down by %10, but this sale price is still marked up from the cost of $270. Find the markup from cost to sale price.
The required markup from the cost of $270 is 20% of the sale price.
The selling price of an item is $360. It is marked down by %10, but this sale price is still marked up from the cost of $270.
What is the percentage?
The percentage is the ratio of the composition of matter to the overall composition of matter multiplied by 100.
here,
Selling price after 10% marked down,
= [1 - 0.10] × 360
= 324
Now,
Percentage marked up of cost price of $270
= 324 - 270 / 270 × 100
= 20%
Thus, as of the given condition markup cost of $270 is 20% of the sale price.
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Can you please help me solve the problem.
Answer: Acute Triangle
Step-by-step explanation:
Because when we look at it we can actually see each of the angles, they are less than 90 degrees then they are consider as acute angles, if it's greater than 90 degrees and smaller than 180 degrees then it will be consider as a obtuse, and if it's equal to 90 degrees then it's a right angle. This is not a equiangular, if it's then it should be have three 60degrees angles.
patricia is studying a polynomial function f(x). three given roots of f(x) are negative 11 minus startroot 2 endroot i, 3 4i, and 10. patricia concludes that f(x) must be a polynomial with degree 4. which statement is true?
so, -11-[tex]\sqrt{2}[/tex]i and 3 + 4i are roots of polynomial function f(x) so 11 + [tex]\sqrt{2}[/tex]i and
3 - 4i are roots of f(x)
Therefore, option D is correct
Polynomial function:
A polynomial function, in general, is also stated as a polynomial or polynomial expression, defined by its degree. The degree of any polynomial is the highest power present in it.
given that
Patricia is studying a polynomial function f(x). three given roots of f(x) are
-11-[tex]\sqrt{2}[/tex]i , 3 + 4i and 10 . Patricia concludes that f(x) must be a polynomial with degree 4.
so -11-[tex]\sqrt{2}[/tex]i and 3 + 4i are roots of polynomial function f(x) so 11 + [tex]\sqrt{2}[/tex]i and
3 - 4i are roots of f(x)
Therefore, option D is correct
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100 POINTS HELP PLEASE WILL GIVE BRAINLEIST An export is a good or service sent out of a country into another country. A factory in the Philippines makes clothing to send to a vendor in Canada. Which country is exporting?
A. both
B. Canada
C. Philippines
Answer:
c.philiippines
Step-by-step explanation:
exporting (present participle)
send (goods or services) to another country for sale:
HELP!!!!! How do I solve these two problems????
The angle x of the isosceles triangle is 64 degrees.
The side and angle of the similar polygon are 7.5 units and 20 degrees.
How to find the angle of a triangle and polygon?The triangle above is an isosceles triangle. An isosceles triangle is a triangle that has 2 sides equal to each other and two angles equal to each other.
Therefore, the base angle of the isosceles triangle are equal.
Hence,
52 + x + x = 180
52 + 2x = 180
2x = 180 - 52
2x = 128
x = 128 / 2
x = 64 degrees
Similar polygons are two polygons with the same shape, but not the same size. Similar polygons have corresponding angles that are congruent, and corresponding sides that are proportional.
Therefore, let's find x and y in the polygon.
x / 5 = 6 / 4
cross multiply
4x = 30
x = 30 / 4
x = 7.5
y - 73 = 360 - 90 - 61 - 116
y - 73 = 93
y = 93 - 73
y = 20
Therefore, the angle x of the isosceles triangle is 64 degrees and the sides x of the polygon is 7.5 units and the angle y of the polygon is 20 degrees.
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3times a number is 4 less than the square of that number find the negative solution
Answer: the answer is -1
What are the coordinates of the circumcenter of a triangle with vertices a(0,1), b(2, 1), and c(2, 5)? enter your answer in the boxes. (, ).
The coordinate of the circumcenter of a triangle is (1,3).
Here we have to find the coordinates of the circumcenter of a triangle.
Given data:
The vertices of the triangle are:
A = (0,1)
B = ( 2,1)
C= ( 2,5)
The formula to find the distance between the two points:
Distance = [tex]\sqrt{(x2 - x1)^{2} + (y2-y1)^{2} }[/tex]
In the same manner distance between AB, BC, and AC are:
AB = [tex]\sqrt{(2-0)^{2} + (1-1)^{2} }[/tex]
= 2
BC = [tex]\sqrt{(2-2)^{2} + (5-1)^{2} }[/tex]
= 4
AC = [tex]\sqrt{(2-0)^{2} + (5-1)^{2} }[/tex]
= 2√5
So by the Pythagoras, we get to know that the triangle is a right-angle triangle.
The circumcenter of a right-angle triangle is the midpoint of the hypotenuse.
Midpoint of AC = (0+2)/2 , (1+5)/2
= 2/2, 6/2
= 1,3
Therefore the coordinate of the circumcenter is (1,3).
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Find the equation of a line that passes through the point (0,5) and has a gradient of 3. Leave your answer in the form y = m x + c.
The equation of a line with point (0,5) and gradient 3 in the form of y = mx + c is y = 3x + 5.
The point slope form of the equation to be used for finding the equation is -
y - [tex] y_{1}[/tex] = m (x - [tex] x_{1}[/tex] )
As per the mentioned information, m or slope is 3, [tex] y_{1}[/tex] is 5 and [tex] x_{1}[/tex] is 0. Keep the values in point slope form to find the equation.
y - 5 = 3 (x - 0)
Performing multiplication on Right Hand Side
y - 5 = 3x - 0
Shifting 5 to Right Hand Side
y = 3x + 5
Thus, the answer is y = 3x + 5.
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When the coordinates 1 1 7 3 8 0 and 2 − 2 are joined which shape is formed 5 points group of answer choices trapezoid rectangle rhombus square?
the coordinates (1,1), (7,3), (8,0) and (2,-2) are joined which shape is formed as Square
When the coordinates (1,1), (7,3), (8,0) and (2,-2) are joined, a square is formed. The sides are equal length and the angles are all 90 degrees.The coordinates (1,1), (7,3), (8,0) and (2,-2) are points on a two-dimensional plane. When the points are connected, a square is formed. A square is a four sided shape with four 90 degree angles and four equal length sides. The length of each side can be calculated by finding the distance between the two points. For example, the distance between (1,1) and (7,3) is 6 units. Therefore, the length of each side of the square is 6 units.
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show that -5(3m-7) and -15+35 are equivalent
Answer:
they are equivalent when m = 1
Step-by-step explanation:
-5(3m-7) = -15+35
-5(3m-7) = 20
-15m+35 = 20
-15m+35 = 20
-15m = -15
m = 1
how many small cube with side length of 6 cm are needed to completely fill a larger cube with side lengths of 30 cm
Answer:
1,728 small cubes are needed to completely fill a larger cube with side lengths of 30 cm.
Step-by-step explanation: