The general equation of the line : y = m * x + b
where m is the slope , b is y -intercept
Given the function :
[tex]y=8x+20[/tex]The rate of change of the equation = the slope of the function
So, by comparing the given equation to the general from
The slope = m = 8
So, the rate of change = 8
A spherical iron ball is coated with a layer of ice of uniform thickness. If the ice melts at the rate of 8 mL/min, how fast is the outer surface area of ice changing when the outer diameter of the ball with ice on it is 24 cm?
When the outer diameter of the ball with ice on it is 24 cm then the outer surface area of ice changes at the rate of 1/90cm²/sec.
As given in the question,
Spherical ball is coated with uniform thickness.
Ice melts at the rate of 8ml/min
= (8/60)cm³/sec
Consider r as the radius of given spherical ball
Diameter = 24cm
⇒Radius 'r' =12cm
Volume 'V' = (4/3)πr³
⇒ dV /dt = (4/3)π(3r²r')
⇒-(8/60) = (4/3)π(3 ×12²×r')
⇒r' =-(8/60)×(1/576π)
⇒r' = - 1/ 4320π
Rate at which change in surface area
A =4πr²
⇒A' = 4πrr'
⇒A' = 4π (12) ( - 1/ 4320π)
= -1/90 cm²/sec.
Decrease in surface area = 1/90 cm²/sec.
Therefore, when the outer diameter of the ball with ice on it is 24 cm then the outer surface area of ice changes at the rate of 1/90cm²/sec.
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which graph represents the solution to -1/2m>7/11
The graph of the inequality:
(-1/2)*m > 7/11
Can be seen in the image at the end.
Which graph represents the solution for the inequality?Here we have the following inequality:
(-1/2)*m > 7/11
First, let's solve this for m, this means that we need to isolate the variable in one side of the inequality.
If we multiply both sides by -2, we get:
-2*(-1/2)*m < -2*(7/11)
Where the direction of the symbol changes because we are multiplying by a negative number.
m < -14/11
The graph of this will be an open circle at -14/11 and an arrow that goes to the left, like the one below.
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An Integer is a number with a fractional part. True or False
An integer is a number with a fractional part is true statement.
You draw 7 cards from a standard deck of cards. What is the probability of drawing 3 diamonds and 2 clubs?
Solution
For this case we can do the following:
[tex]p=\frac{\text{possible}}{\text{total}}[/tex]and we can find the answer with this:
[tex]p=\frac{(13C3)(13C2)(26C2)}{52C7}=0.0541[/tex]We are reviewing a module and I don't remember how to do it.
The coordinate of point X which is 5/6 of the distance between P and Q is 5/11
Here, we want to calculate the coordinates of point X which is 5/6 of the distance between P and Q
Mathematically, we can use the internal division formula.
In this case, the coordinates of y is 0 in all cases
So the coordinates of P is (-5,0) while the coordinates of Q is (7,0)
Now, the coordinates of X divides the line PQ in the ratio 5 to 6
Using the internal divison formula, we have;
[tex](x,y)\text{ = }\frac{mx_2+nx_1}{m+\text{ n}},\text{ }\frac{my_2+ny_1}{m+\text{ n}}[/tex]In this case however, we are going to focus on the x-axis part of the question since the values of y at all points is 0
m , n are the division values which are 5 and 6 respectively in this case
x2 is 7 while x1 is -5
Substituting all of these, we have;
[tex]\begin{gathered} (x,y)\text{ = }\frac{5(7)\text{ + 6(-5)}}{11},\text{ 0} \\ \\ (x,y)\text{ = }\frac{35-30}{11},\text{ 0} \\ (x,y)\text{ = }\frac{5}{11},\text{ 0} \end{gathered}[/tex]So the coordinate of point X which is 5/6 of the distance between P and Q is 5/11
I need help answering the questions for person 2 on my group assignment
The equation for the relation of sides of triangle can be obtained by similar triangle property.
Consider triangle ABC and triangle DBE.
[tex]\begin{gathered} \angle CAB=\angle EDA\text{ (Each angle is right angle)} \\ \angle CBA=\angle EBD\text{ (common angle)} \\ \Delta CBA\cong\Delta EBD\text{ (By AA similarity condition)} \end{gathered}[/tex]Determine the ratio of corresponding sides of simillar triangle.
[tex]\frac{CB}{EB}=\frac{BA}{BD}=\frac{CA}{ED}[/tex]Thus similar triangle property is used to set up the equation.
Options for this are: 20 of the best selling cameras, same photographer, 100 pictures with each camera, consistent across all cameras 10 point scale, two were from companies who are major advertisers
It is given that:
A writer for a magazine recently did a test to determine which mid-range digital camera takes the best pictures. Her method is described below.
Which part of the method describes an area of potential bias?
She gathered 20 of the best.selling cameras and used the same photographer to take 100 pictures with each camera .She ensured that the environment and the subject of each picture were consistent across all cameras and used a 10.point scale to determine picture quality. Of the cameras tested, two were from companies who are major advertisers in the magazine.
Now if the reading is done carefully, it can be concluded that the information given by:
"Of the cameras tested, two were from companies who are major advertisers in the magazine." can be considered for a potential bias since the magazine may be pressured by these two companies to give them a higher rating than they deserve.
So the option:two were from companies who are major advertisers is correct.
If the correlation coefficient is 1, then the relation is a __________________.perfect positive correlationperfect negative correlationweak negative correlationweak positive correlation
Given:
The correlation coefficient is 1.
Required:
What type of correlation is it?
Explanation:
A coefficient of -1.0 indicates a perfect negative correlation, and a correlation of 1.0 indicates a perfect positive correlation.
Answer:
Hence, correlation coefficient is 1 then relation is perfect positive correlation.
Can you please help me with 44Please use all 3 forms of the expression such as : down/up. As _,_ And limits
Answer:
[tex]\begin{gathered} x-\text{intercept}=-3\text{ and 1} \\ y-\text{intercept}=\text{ -9} \end{gathered}[/tex][tex]\begin{gathered} x=1\text{ multiplicity 3} \\ x=-3\text{ multiplicity 2} \\ \lim _{x\rightarrow\infty}(x-1)^3(x+3)^2=\infty \\ \lim _{x\rightarrow-\infty}(x-1)^3(x+3)^2=-\infty \end{gathered}[/tex]Step-by-step explanation:
To find the x-intercepts factor the function to the simplest form:
[tex]h(x)=(x-1)^3(x+3)^2[/tex]As we can see the zeros to the function would be 1 and -3, then its:
[tex]\begin{gathered} x-\text{intercept}=-3\text{ and 1} \\ y-\text{intercept}=\text{ -9} \end{gathered}[/tex]Zero has a "multiplicity", which refers to the number of times that its associated factor appears in the polynomial. Therefore, this function has multiplicity:
[tex]\begin{gathered} x=1\text{ multiplicity 3} \\ x=-3\text{ multiplicity 2} \end{gathered}[/tex]For the end behavior:
down/up
As x approaches infinity f(x) approaches infinity
As x approaches -infinity f(x) approaches -infinity
[tex]\begin{gathered} \lim _{x\rightarrow\infty}(x-1)^3(x+3)^2=\infty \\ \lim _{x\rightarrow-\infty}(x-1)^3(x+3)^2=-\infty \end{gathered}[/tex]In which month was the average temperature closest to 0°C ?
Find all values for which at least one denominator is equal to 0.
Given:
There are given the expression:
[tex]\frac{4}{x+2}-\frac{5}{x}=1[/tex]Explanation:
To find the value of x that is equal to 0, we need to perform LCM in the denominator and then find the value for x:
Then,
From the given expression:
[tex]\begin{gathered} \frac{4}{x+2}-\frac{5}{x}=1 \\ \frac{4x-5(x+2)}{x(x+2)}=1 \end{gathered}[/tex]Then,
According to the question, the values at least one denominator is equal to .
So,
[tex]\begin{gathered} x(x+2)=0 \\ x=0 \\ x+2=0 \\ x=-2 \end{gathered}[/tex]Final answer:
Hence, the value of x is shown below:
[tex]x\ne0,-2[/tex]
Figure R = figure R". Describe a sequence of three transformations you canperform on figure R to show this. Show your work.
1) Rotate the figure 90º clockwise.
To rotate a figure 90º clockwise you have to perform the following transformation:
(x,y)→(y,-x)
So for each point of the figure you have to swich places between x and y.
And you have to multiply the x coordinate by -1.
R has 6 points:
(-4,-5)
(-7,-5)
(-7,-4)
(-5,-4)
(-6,-3)
(-2,-2)
First step: swich places between the coordinates of each point:
(x,y) → (y,x)
(-4,-5)→(-5,-4)
(-7,-5)→(-5,-7)
(-7,-4)→(-4,-7)
(-5,-4)→(-4,-5)
(-6,-3)→(-3,-6)
(-2,-2)→(-2,-2)
Second step, multiply the y-coordinate of the new set by -1
(y,x)→ (y,-x)
(-5,-4)→ (-5,4)
(-5,-7)→ (-5,7)
(-4,-7)→ (-4,7)
(-4,-5)→ (-4,5)
(-3,-6)→ (-3,6)
(-2,-2)→ (-2,2)
After the rotation, the figure moved from the 3rth quadrant to the 2nd quadrant.
plant A produced 3 times as many panels as plant b. two percent of the panels from plant A and 5% of the panels from plant b were defective. how many panels did plant b produce if the two plants together produced 990 defective panels
let x the number of panels that Plant A produced
y the number of panels that Plant B produced
then, we have
x = 3y
0.02x + 0.05y = 990
and solve the system:
[tex]\begin{gathered} 0.02(3y)+0.05y=990 \\ 0.06y+0.05y=990 \\ 0.11y=990 \\ \frac{0.11y}{0.11}=\frac{990}{0.11} \\ y=9000 \end{gathered}[/tex]answer: plant b produced 9000 panels
Based on the information given, can you determine that the quadrilateral must be a parallelogram? Explain. Given: XN NZ and NY = NW No; you cannot determine that the quadrilateral is a parallelogram. Yes; opposite sides are congruent. Yes; two opposite sides are both parallel and congruent. Yes; diagonals of a parallelogram bisect each other.
A parallelogram is a quadrilateral that has the following characteristics:
The opposite sides are parallel and congruent.
The opposite angles are congruent.
The consecutive angles are supplementary.
If any one of the angles is a right angle, then all the other angles will be at right angle.
The two diagonals bisect each other.
Since:
[tex]XN\cong NZ;NY\cong NW[/tex]We can conclude that the answer is:
Yes; diagonals of a parallelogram bisect each other.
What is the domain, range, and function? {(-3, 3), (1, 1), (0, -2), (1,4), (5, -1)}
The domain are all the inputs, that is; the x-values
Domain = { -3, 0, 1, 5}
The range are all the output, that is the y-values
Range = { 3, 1, -2, 4, -1}
This is just a relation and not a function, as we have more than 2 same value of x
A woman is floating in a
boat that is 175 feet from
the base of a cliff. The cliff
is 250 feet tall. What is the
angle of elevation from
the boat to the top of the
cliff?
The angle of depression between the cliff and the boat is 55.0
What is angle of depression?
The angle of depression is the angle between the horizontal line and the observation of the object of from the horizontal line. It's basically used to get the of distance of the two objects where the angles and an of object's distance from the ground are known to us.
A boat is moving 175 feet from the base and a women is in the boat.the height of the cliff is 259 feet tall. Here we have to find the angle between the cliff and the boat.
As per the given question
We have a right angled traingle where base is 175 ft and height is 250 ft.
Thus,
We know that tan theta =opposite/adjacent
250/175
So theta=tan^-1(250/175)
So theta = 55.0
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This chart shows the cost per pound of different fruits.
From the given data, the cost per pound of apple, CP=$1.89≈$2.
We have to estimate the cost of n=3.2≈3 pounds(lb) of apple.
The cost of n=3 lb of apple can be calculated as,
[tex]\begin{gathered} T=CP\times n \\ =2\times3\text{ lb} \\ =6 \end{gathered}[/tex]Therefore, the cost of 3. pounds of apples is about $6.048.
Find an equation of the line. Write the equation using function notation.
Through (4, -1); perpendicular to 4y=x-8
The equation of the line is f(x) =
The equation of line perpendicular to 4y = x-8 passing through (4,-1) is:
[tex]y = -4x+15[/tex].
What is a equation of line?These lines are written in the form y = mx + b, where m is the slope and b is the y-intercept. We know from the question that our slope is 3 and our y-intercept is –5, so plugging these values in we get the equation of our line to be y = 3x – 5.
Given equation of line is:
4y=x-8
We have to convert the given line in slope-intercept form to find the slope of the line
Dividing both sides by 4.
[tex]y = \frac{1}{4}x-2[/tex]
Let [tex]m_{1}[/tex] be the slope of given line
Then,
[tex]m_{1}[/tex] = [tex]\frac{1}{4}[/tex]
Let [tex]m_{2}[/tex] be the slope of line perpendicular to given line
As we know that product of slopes of two perpendicular lines is -1.
[tex]m_{1}*m_{2} = -1\\\frac{1}{4}*m_{2}=-1\\ m_{2} = -4[/tex]
The slope intercept form of line is given by
[tex]y = m_{2}x+c[/tex]
[tex]y = -4x+c[/tex]
to find the value of c, putting (4,-1) in equation
[tex]-1 = -4*4+c\\-1+16 = c\\c = 15[/tex]
Putting the value of c in the equation
[tex]y=-4x+15[/tex]
Hence, The equation of line perpendicular to 4y = x-8 passing through (4,-1) is [tex]y = -4x+15[/tex].
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Jaylen used a 20% discount on a pair of jeans that cost $70 before tax. The sales tax is 6%. How much does the pair of jeans cost after tax? Show your work
If Jaylen used a 20% discount on a pair of jeans that cost $70 before tax, then the price of the jeans will be $70 - 20%.
Let's calculate the 20% of $70.
[tex]70\times20\%=14[/tex]So, the discounted price of the jeans before tax is $70 - $14 = $56.
Generally, sales tax is applied to the discounted price, so let's calculate the 6% of $56.
[tex]6\%\times56=3.36[/tex]The sales tax is $3.36.
Therefore, the cost of the pair of jeans after tax is $59.36.
[tex]56+3.36=59.36[/tex]The length of a rectangle is 6 cm more than the width. If the perimeter is 52 cm. What are the dimensions of the rectangle?
LA rectangle has two pairs of sides of the same length. If we call W to the width of the rectangle, we know that the length is 6cm more. If we call L the length of the rectangle:
[tex]L=W+6[/tex]The perimeter of a rectangle is twice the length plus twice the width:
[tex]Perimeter=2L+2W[/tex]Since we know that the perimeter is 52 cm, we can write the system of equations:
[tex]\begin{cases}L={W+6} \\ 2L+2W=52{}\end{cases}[/tex]We can substitute the first equation into the second one:
[tex]2(W+6)+2W=52[/tex]And solve:
[tex]2W+12+2W=52[/tex][tex]\begin{gathered} 4W=52-12 \\ . \\ W=\frac{40}{4}=10\text{ }cm \end{gathered}[/tex]We know that W = 10cm, we can now find L:
[tex]L=10+6=16\text{ }cm[/tex]Thus, the dimensions of the rectangle are:
Length: 16 cm
Width: 10 cm
The area of a picture projected on a wall varies directly at the square of the distance from the projector to the wall if a 10ft distance produces a 16 feet squared (^2) picture, what is the area of the picture produced when the projection unit is moved to a distance 20 ft from the wall?
The new picture is 64 ft squared. Surface area refers to the area of an open surface or the boundary of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a form or planar lamina.
What is area?The measurement that expresses the size of a region on a plane or curved surface is called area. Surface area refers to the area of an open surface or the boundary of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a form or planar lamina.
We are given the relation: Area of pic = constant * d^2, where d is distance from projector to wall.
For d = 10, we have A = 16 ft sqrd
Now given d = 20
what is A?
constant = 16/10*10
new A = [16/100] * 20*20 = 16 * 4 = 64 ft sqrd.
The new picture is 64 ft squared.
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Which of these steps will eliminate a variable in this system?3x-3y=66x+9y=3OA. Multiply the first equation by 3. Then subtract the second equationfrom the first.B. Multiply the first equation by 2. Then add the equations.C. Multiply the first equation by 2. Then subtract the second equationfrom the first.OD. Multiply the second equation by 2. Then subtract the secondequation from the first.
The given system of equation is:
[tex]\begin{gathered} 3x-3y=6 \\ 6x+9y=3 \end{gathered}[/tex]Multiply through the first equation by 2:
[tex]\begin{gathered} 6x-6y=12 \\ 6x+9y=3 \end{gathered}[/tex]Subtract the second equation from the first equation to get:
[tex]-15y=9[/tex]Therefore, the steps that will eliminate the variable x are:
Multiply the first equation by 2. Then subtract the second equation from the first.
Choice C
How many and what type of solution(s) does the equation have?6p2 = 8p + 32 rational solutions1 rational solutionNo real solutions2 irrational solutions
We are going to solve the question using the quadratic formula
[tex]\begin{gathered} x=\frac{-b\pm\sqrt[]{(b^2}-4ac)}{2a} \\ \text{where the quadratic equation is ax}^2+bx+c=0 \end{gathered}[/tex]The quadratic equation given is
[tex]\begin{gathered} 6p^2=8p+3 \\ 6p^2-8p-3=0 \\ \text{where a=6} \\ b=-8 \\ c=-3 \end{gathered}[/tex]By substitution we will have,
[tex]\begin{gathered} p=\frac{-(-8)\pm\sqrt[]{(-8)^2}-(4\times6\times-3)}{2\times6} \\ p=\frac{8\pm\sqrt[]{64+72}}{12} \\ p=\frac{8\pm\sqrt[]{136}}{12} \\ p=\frac{8\pm\sqrt[]{4\times34}}{12} \\ p=\frac{8\pm2\sqrt[]{34}}{12} \\ p=\frac{2(4\pm\sqrt[]{34)}}{12} \\ p=\frac{4\pm\sqrt[]{34}}{6} \\ p=\frac{4+\sqrt[]{34}}{6}\text{ or p=}\frac{4-\sqrt[]{34}}{6} \end{gathered}[/tex]Therefore,
With the roots gotten from the quadratic equation, we can therefore deduce that the solutions to the equation 6p²=8p+3 will give 2 irrational roots.
The correct answer is OPTION D
Evaluate f(2) and f(2.1) and use the results to approximate f '(2). (Round your answer to one decimal place.)f(x) = x(9 − x)f '(2) ≈
Given a function f(x) = x(9 - x).
We need to find the value of f(2) and f(2.1) and use them to approximate the value of f'(2).
The value of f(2) is calculated below:
[tex]\begin{gathered} f(2)=2(9-2) \\ =2(7) \\ =14 \end{gathered}[/tex]The value of the f(2.1) is calculated as follows:
[tex]\begin{gathered} f(2.1)=2.1(9-2.1) \\ =2.1(6.9) \\ =14.49 \end{gathered}[/tex]Now, by the definition of f'(x), we know that
[tex]f^{\prime}(x)=\frac{f(x+\Delta x)-f(x)}{(x+\Delta x)-x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}[/tex]For the given condition, x = 2, and delta x = 0.1. So, the value of f'(2) is
[tex]\begin{gathered} f^{\prime}(2)=\frac{f(2+0.1)-f(2)}{0.1} \\ =\frac{f(2.1)-f(2)}{0.1} \\ =\frac{14.49-14}{0.1} \\ =\frac{0.49}{0.1} \\ =4.9 \end{gathered}[/tex]Thus, the approximate value of f'(2) is 4.9.
Identify the property of real numbers illustrated in the following equation.(-5) + (y · 7) = (y · 7) + (-5)
By definition, the commutative property of addition says that changing the order of addends does not change the sum, which is precisely what the equation is trying to show by changing the order of the sum, therefore, the property illustrated is the commutative property of addition.
Julian is decorating the outside of a box in the shape of a right rectangular prism. Thefigure below shows a net for the box.
The surface area of the box equals the sum of the surface area of each of its parts.
And the area of each rectangle that form the box is found by multiplying the width by the height of that rectangle.
We have two ractangles with sides 7 ft and 10 ft. So the area of each one is:
7 ft * 10 ft = 7 * 10 * ft * ft = 70 ft²
Since there's two of this rectangle, their areas sum up to
2 * 70 ft² = 140 ft²
Now, we also have two rectangles with sides 7 ft and 14 ft (the second and the fourth rectangles from left to right in the image). So, their areas sum up to:
2 * (7 ft * 14 ft) = 2 * (98 ft²) = 196 ft²
Finally, we also have two rectangles with sides 10 ft and 14 ft. Then, their area together is:
2 * (10 ft * 14 ft) = 2 * (140 ft²) = 280 ft²
Therefore the total surface area of the box is the sum:
140 ft² + 196 ft² + 280 ft² = 616 ft²
rounded 425.652 to the hundredths place
Since the given number is 425.652
The hundredth digit is the 2nd number right at the decimal point
It is 5
To round to the nearest hundredth, we will look at the digit right to it
1. If it is 0, 1, 2, 3, or 4 we will ignore it and write the number without change except by canceling that digit
2. If it is 5, 6, 7, 8, or 9 we will cancel it and add the digit left to it 1
Since the right digit to the digit 5 is 2, then we will remove it and do not change the digit 5 (case 1), then
The number after rounding should be 425.65
The answer is 425.65
last night Danielle had a birthday party. 1/3 of the cake was left over.She wanted to share the left over cake with 4 friends the next day .How much of birthday cake would each get
Solution
For this case we have a total cake representing 1
We also know that 1/3 of the cake was left over so then 1/3
And we want to share to 4 friends so we can do this:
1/3 * 1/4 = 1/12
So then each friend will recieve 1/12
Find a polynomial function of lowest degree with rational coefficients that has the
given numbers as some of its zeros.
√3,51
The polynomial function in expanded form is f(x) =
Answer: [tex]f(x)=x^3 -51x^2 -3x+153[/tex]
Step-by-step explanation:
By the conjugate root theorem, the roots are [tex]\sqrt{3}, -\sqrt{3}, 51[/tex].
Letting the leading coefficient be 1,
[tex]f(x)=(x-\sqrt{3})(x+\sqrt{3})(x-51)\\\\=(x^2 -3)(x-51)\\\\=x^3 -51x^2 -3x+153[/tex]
What is 4x+10(2x) - 8x
4x+10(2x) - 8x
First, multiply to solve the parentheses:
4x+20x-8x
Add and subtract
16x
