The formula for the volume of a sphere is given by:
[tex]V=\frac{4}{3}\pi r^3[/tex]Since we are asked the rate of change of the volume with repect to time, we take the derivative on both sides, taking into account the chain rule:
[tex]\frac{dV}{dt}=\frac{d(\frac{4}{3}\pi r^3)}{dt}[/tex]taking out the constants:
[tex]\frac{dV}{dt}=\frac{4}{3}\pi\frac{d(r^3)}{dx}[/tex]Now we derivate, using the chain rule, that is:
[tex]\frac{df(g(x))}{dx}=f^{\prime}(x)g^{\prime}(x)[/tex]Applying the rule:
[tex]\frac{dV}{dt}=\frac{4}{3}\pi(3r^2)\frac{dr}{dt}[/tex]Simplifying:
[tex]\frac{dV}{dt}=4\pi(r^2)\frac{dr}{dt}[/tex]We have the following known values:
[tex]\begin{gathered} \frac{dr}{dt}=\frac{2\operatorname{cm}}{s} \\ r=3\operatorname{cm} \end{gathered}[/tex]Replacing we get:
[tex]\frac{dV}{dt}=4\pi(3)^2(2)[/tex]Solving we get:
[tex]undefined[/tex]A box has s snack bags in it. Each snack bag contains c carrot sticks.Which equation can be used to find b , the number of carrot sticks in one box?Ab = s/cB b = scCb = s+c Db = c/s
Solution
For this case we have s snacks and each snack with c carrot sticks
So then if we want to find the total of carrot sticks in one box we can do the following operation:
B. b = sc
Reason quantitatively. The two rectangles shown
are similar. What is the value of x
Two shapes are similar if the ratio of the lengths of their corresponding sides are equal.
Both shapes given in the question are rectangles, therefore, one pair of opposite sides is longer than the other.
We can find the ratio for the bigger rectangle since it has all the values complete and then compare this ratio to the smaller rectangle to find the value of the unknown side.
The ratio of the longer side to the shorter side for the bigger rectangle is
[tex]\begin{gathered} \frac{16}{2} \\ =8 \end{gathered}[/tex]Therefore, for the smaller rectangle, the ratio of the longer side to the shorter side is
[tex]\frac{4}{x}=8[/tex]Solving for x, we have
[tex]\begin{gathered} x=\frac{4}{8} \\ x=0.5 \end{gathered}[/tex]The value for x is 0.5.
i got question A &B i just can’t get C
Step 1
The domain refers to all values that go into a function. The range refers to all the values that come out.
Step 2
Find the domain
[tex]D=\mleft\lbrace2007,2008,2009\mright\rbrace[/tex]Step 3
Find the range
[tex]R=\mleft\lbrace234300,213200,\text{ 212,200}\mright\rbrace[/tex]Determine if the shape is a polyhedron using Eulers formula choices: 8, 12, 3, 5,9, 2, 18
Given:
The given shape is a polyhedron.
Required:
We need to use Euler's formula for the given shape.
Explanation:
Consider Euler's formula.
[tex]V-E+F=2[/tex]where V is the number of vertices, E is the number of edges and F is the number of faces.
Recall that a vertex is a corner.
The number of corners in the given shape is 12.
[tex]V=12[/tex]Recall that an edge is a line segment between faces and a face is a single flat surface.
The number of edges in the given shape is 18.
[tex]E=18[/tex]The number of faces in the given shape is 8.
[tex]F=8[/tex][tex]V-E+F=12-18+8[/tex][tex]V-E+F=2[/tex]The given shape satisfied Euler's formula.
So it is a polyhedron.
Final answer:
[tex]V=12[/tex]
[tex]E=18[/tex]
[tex]F=8[/tex]
[tex]V-E+F=2[/tex]
The given shape is a polyhedron.
Find the number of degrees in the acute angle formed by the intersection of walnut street and elm street
Given two parallel lines and a transversal
So, the angles (2x + 33) and ( 5x - 15 ) are congruent because they are corresponding angles
So, 5x - 15 = 2x + 33
Solve to find x
[tex]5x-15=2x+33[/tex]Combine like terms:
[tex]\begin{gathered} 5x-2x=33+15 \\ 3x=48 \\ x=\frac{48}{3}=16 \end{gathered}[/tex]So, the required angle = 2x + 33 = 2 * 16 + 33 = 32 + 33 = 65
So, the angle is 65
Use the graph to complete the statements.For every dollar you spend, you can getpounds of grapes.For each pound of grapes, you would need $
We are given a graph of pounds of grapes vs dollar spent.
To know what $1 worth of grapes is, we go to the horizontal axis to locate 1 and trace up to where it meets up with the graph and on getting that point on the graph, we trace it left to the vertical axis to get $0.5 (50 cent)
To get the dollars' worth for each grape, we locate one on the vertical axis and trace it right to where it cuts the graph. This can easily be seen to
By what factors could each equation be multiplied in order to solve the system by linear combination? 2x-3y=8 and 5x+4y=-3A. First equation by 2: second equation by 5 B. First equation by 3; second equation by 4 C. First equation by 3: second equation by 8 D. First equation by 4: second equation by 3
we have these equations
[tex]\begin{gathered} 2x-3y=8 \\ 5x+4y=-3 \end{gathered}[/tex]In order to solve this system for the x variable, we can multiply the first equation by 4 and the second by 3
This correspond to answer D.
A new cell tower is being constructed and needs a guy-wire connected 137 feet up the tower and it needs to make an angle of 56° with the ground. What length does the wire need to be?
2) 58, 67, 44, 72, 51, 42, 60, 46, 69Minimum :Maximum :Q,Q2:Q,
Given the following data set:
58, 67, 44, 72, 51, 42, 60, 46, 69
First, we will arrange the data in order from the least to the greatest.
42, 44, 46, 51, 58, 60, 67, 69, 72
The minimum = 42
The maximum = 72
Q2 = the median of the data = the number that in the middle
As the set has 9 data, so, the median will be the data number 4
Q2 = 58
To find Q1 and Q3 , the data will be divided into two equal groups
(42, 44, 46, 51), 58, (60, 67, 69, 72)
Q1 = the median of the first group = (44+46)/2 = 45
Q3 = the median of the second group = (67+69)/2 = 68
So, the answer will be:
Minimum : 42
Maximum : 72
Q1 : 45
Q2 : 58
Q3 : 68
What is the constant of proportionality of the tablex 5 8 11y 35 56 77
y =kx
Where:
k = Constant of proportionality
If x = 5, y = 35
35 = k5
Solving for k:
k = 35/5 = 7
Verify the answer:
If x = 8 , y = 56
y = kx = 7*8 = 56
The constant of proportionality is 7
Find the equation of a line that contains the point (-2, -6) is perpendicular to the line
Answer:
answer Is y= 5x+4......
m varies directly with n. Determine m when n=8 and k= 16
We have that m varies directly with n, then:
[tex]m=kn[/tex]now, if n =8 and k=16, then:
[tex]\begin{gathered} m=(16)(8)=128 \\ m=128 \end{gathered}[/tex]therefore, m = 128
A file that is 284 megabytes is being downloaded. If the download Is 17.5% complete, how many megabytes have been downloaded? Round your answer to thenearest tenth.megabytesх5?
We are given the size of a file to download is 284 megabytes. If 17.5% has complete,
We want to find the megabytes that has been downloaded
Solution
We have
[tex]284mb[/tex]The 17.5% of 284mb will be
[tex]\begin{gathered} \frac{17.5}{100}\times284 \\ =\frac{497}{10} \\ =49.7mb \end{gathered}[/tex]Therefore, the megabytes that have been downloaded is 49.7mb (to the nearest tenth)
How would I figure out 4 could you draw it out so I could understand betters it’s my first day learning this
Question 4
The sketch of the isosceles right triangle is given below
For an isosceles right triangle, the two legs are equal
So we will get the value x as follow
[tex]\begin{gathered} x^2+x^2=8^2 \\ 2x^2=8^2 \\ 2x^2=64 \\ x^2=32 \\ x=4\sqrt[]{2} \end{gathered}[/tex]The perimeter of the triangle can be obtained as follow
The perimeter is simply the sum of all the sides of the triangle
[tex]\begin{gathered} \text{Perimeter}=x+x+8 \\ \text{Perimeter}=4\sqrt[]{2}+4\sqrt[]{2}+8=8\sqrt[]{2}+8 \\ \text{Perimeter}=8\sqrt[]{2}+8 \\ \text{Perimeter}=8(\sqrt[]{2}+1) \\ \text{Perimeter}=19.31\text{ units} \end{gathered}[/tex]To get the area of the triangle
we will use the formula
[tex]\begin{gathered} \text{Area}=\frac{1}{2}\times base\times\text{height} \\ \text{Area}=\frac{1}{2}\times4\sqrt[]{2}\times4\sqrt[]{2} \\ \text{Area}=2\sqrt[]{2}\times4\sqrt[]{2} \\ \text{Area}=2\times4\times2 \\ \text{Area}=16 \end{gathered}[/tex]The area of the triangle is 16 square units
Reflection over the y-axis Example 2 Original Point Coordinates Image Point Coordinates A (-8,2) A B (-4,9) B C (-3,2) C'
We have to reflect the 3 points shown over the y-axis.
The simple rule for reflecting over y-axis:
• keep y coordinate same
,• negate the x coordinate
So,
(x,y) would become (-x,y)
Now, let's reflect the 3 points:
A(-8,2) would become A'(8,2)
B(-4,9) would become B'(4,9)
C(-3,2) would become C'(3,2)
I need help with my math
Answer:
Jeff's popcorn container will hold more popcorn
The bigger container will hold 130 cubic cm more popcorn than the smaller container
Jeff's popcorn container has the following measurement
20.5cm x 10cm x 10cm
George's container has the following measurement
30cm by 8cm by 8cm
Volume of each container can be calculated as = Length x width x height
Volume of Jeff's container = 20.5 x 10 x 10 = 2, 050 cubic cm
Volume of George's container = 30 x 8 x 8
Volume of George's container = 1, 920 cubic cm
Therefore, Jeff's popcorn container will hold more popcorn
The bigger container = 2,050 cubic cm
The smaller container = 1, 920
The amount of popcorn the bigger container can hold more = 2050 - 1920
= 130 cubic cm
Therefore, the bigger container can hold 130 cubic cm more popcorn than the smaller container.
Let n =2. Evaluate the following (nn)n
We have the following:
[tex](nn)n[/tex]n=2
[tex]2\cdot2\cdot2=8[/tex]Let f(-1)=16 and f(5) = -8a. Find the distance between these pointsb. Find the midpoint between these pointsc. Find the slope between these points
We are given the following information
f(-1) = 16 and f(5) = -8
Which means that
[tex](x_1,y_1)=(-1,16)\text{and}(x_2,y_2)=(5,-8)[/tex]a. Find the distance between these points
Recall that the distance formula is given by
[tex]d=\sqrt[]{\mleft({x_2-x_1}\mright)^2+\mleft({y_2-y_1}\mright)^2}[/tex]Let us substitute the given points into the above distance formula
[tex]\begin{gathered} d=\sqrt[]{({5_{}-(-1)})^2+({-8_{}-16_{}})^2} \\ d=\sqrt[]{({5_{}+1})^2+({-24_{}})^2} \\ d=\sqrt[]{({6})^2+({-24_{}})^2} \\ d=\sqrt[]{36^{}+576^{}} \\ d=\sqrt[]{612} \end{gathered}[/tex]Therefore, the distance between these points is √612 = 24.738
b. Find the midpoint between these points
Recall that the midpoint formula is given by
[tex](x_m,y_m)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]Let us substitute the given points into the above midpoint formula
[tex]\begin{gathered} (x_m,y_m)=(\frac{-1_{}+5_{}}{2},\frac{16_{}+(-8)_{}}{2}) \\ (x_m,y_m)=(\frac{-1_{}+5}{2},\frac{16_{}-8}{2}) \\ (x_m,y_m)=(\frac{4}{2},\frac{8}{2}) \\ (x_m,y_m)=(2,4) \end{gathered}[/tex]Therefore, the midpoint of these points is (2, 4)
c. Find the slope between these points
Recall that the slope is given by
[tex]m=\frac{y_2−y_1}{ x_2−x_1}[/tex]Let us substitute the given points into the above slope formula
[tex]m=\frac{-8-16}{5-(-1)}=\frac{-24}{5+1}=\frac{-24}{6}=-4[/tex]Therefore, the slope of these points is -4.
7.4.PS-13 Question Help David drew this diagram of a picture frame he is going to make. Each square represents 1 square inch. What is the area of the picture frame? 12- 10- 0 2 4 6 8 10 12 14 16 18 The area is Enter your answer in the answer box and then click Check Answer. Clear All Check Ans All parts showing of 10 Next → Back Question 7 Review progress
52
1) In this question, since we need to calculate the shaded region or the frame. We'll calculate the whole picture, and then subtract the white rectangle from it.
2) Examining the picture, we can see that the whole larger shape has a width of
14 -4 = 10 units Horizontal (width)
7-1 = 6 units Vertical (height)
3) Let`s use now the formula for the Rectangle Area
[tex]\begin{gathered} A_{\text{Whole Rectangle}}=w\cdot l \\ A_{\text{Whole Rectangle}}=6\times10=60units^2 \\ A_{\text{White Rectangle}}=2\times4=8u^2 \\ A_{\text{FRAME}}=60-8=52units^2 \end{gathered}[/tex]Hence the area of the frame is 52 square units.
If you are selling your house with a local realtor who requires a 5 Pete cent commission fee what can you expect to pay the realtor of your house sells for 170,000
2. Find the values of x, y, and z. The diagram is not to scale.A. x = 85, y = 95, z = 74B. x = 74, y = 85, z = 95C. x = 74, y = 95, z = 85D. x = 85, y = 74, z = 95
Answer:
D. x = 85, y = 74, z = 95
Explanation
The sum of interior angles in the big trangle is 180degrees. Hence;
38 + 57 + x = 180
95 +x = 180
x = 180 - 95
x = 85degrees
The angle x and z are also supplementary since they bith lie on the same stright line. Hence;
85 +
Similarly, the sum of angle in the smaller triangle is 180degrees hence;
11 + z + y = 180
11 + 95 + y = 180
106 + y = 180
y = 180 - 106
y = 74degrees
Hence the value of x, y and z are 85, 74 and 95 degrees respectively
how do you identify sets of real numbers?
The set of number that best describe each situation is shown below:
[tex]\begin{gathered} \text{Whole numbers: These are natural counting positve numbers. e.g 1,2,3,4,5,etc} \\ \text{Integers: These are whole numbers that are positive, negative and zero. e.g 0,1,-1,2,-2,etc} \\ Rational\text{ numbers: These are numbers that can be expressed in the form of }\frac{a}{b},\text{ where b}\ne0,1.\text{ e.g 1/2, 3/5 etc} \\ \text{Irrational numbers: These are numbers that can be expressed in the form of }\sqrt[]{p}\text{ wh}ere\text{ p is prime. e.g }\sqrt[]{2,}\text{ }\sqrt[]{3}\text{ etc} \end{gathered}[/tex]Real numbers in general are majorly sub-divided into two(2) and they are Rational and Irrational numbers.
What does y equal? -y=5y-6
Subtract 5y from both sides of the equation:
[tex]-y-5y=5y-5y-6[/tex][tex]-6y=-6[/tex]Divide both sides by -6
[tex]-\frac{6y}{-6}=-\frac{6}{-6}[/tex][tex]y=1[/tex]Hi I need help with these problems only 1 and 3 since my teacher told us to do even number and if I don't know what to do at all
Since we are dealing with a right triangle, we can use the following trigonometric identities
[tex]\sin \theta=\frac{O}{H},\cos \theta=\frac{A}{H}[/tex]Where θ is an inner angle (different than 90°) of the triangle, O is the opposite side to θ, A is the adjacent side to θ, and H is the hypotenuse.
a) In our case,
[tex]\begin{gathered} \theta=30\text{degre}e \\ H=14,A=m,O=n \\ \Rightarrow\sin (30degree)=\frac{n}{14} \\ \Rightarrow n=14\cdot\sin (30degree)=14\cdot0.5=7 \\ \Rightarrow n=7 \end{gathered}[/tex]and
[tex]\begin{gathered} \Rightarrow\cos (30degree)=\frac{m}{14} \\ \Rightarrow m=14(\cos (30degree))=14\cdot\frac{\sqrt[]{3}}{2}=7\sqrt[]{3} \\ \Rightarrow m=7\sqrt[]{3} \end{gathered}[/tex]The answers are n=7 and m=7sqrt(3).
3) In a diagram, the problem states
Using the same trigonometric identities mentioned in part 1) (plus the tangent function), we get
[tex]\begin{gathered} \sin (30degree)=\frac{18}{H},\tan (30degree)=\frac{18}{A} \\ \Rightarrow H=\frac{18}{\sin(30degree)},A=\frac{18}{\tan(30degree)}=\frac{18}{\frac{1}{\sqrt[]{3}}}=18\sqrt[]{3} \\ \Rightarrow H=\frac{18}{0.5}=36,A=18\sqrt[]{3} \\ \Rightarrow H=36,A=18\sqrt[]{3} \end{gathered}[/tex]The hypotenuse is equal to 36 ft, and the other leg is equal to 18sqrt(3) ft
A right rectangular prism has length 3 3 ft, width 13 ft, and height 23 ft. 1 You use cubes with fractional edge length ft to find the volume. How many cubes are there for each 3 of the length, width, and height of the prism? Find the volume. How many cubes are there for each of the length, width, and height of the prism? cubes, the width has The length has cubes, and the height has cubes.
First, we need to convert the mixed numbers into fractions
[tex]3\frac{1}{3}=\frac{3\cdot3+1}{3}=\frac{10}{3}\text{ ft}[/tex][tex]1\frac{1}{3}=\frac{1\cdot3+1}{3}=\frac{4}{3}\text{ ft}[/tex][tex]2\frac{1}{3}=\frac{2\cdot3+1}{3}=\frac{7}{3}\text{ ft}[/tex]To find how many cubes fit on the length, we need to divide 10/3 by 1/3, as follows:
[tex]\frac{\frac{10}{3}}{\frac{1}{3}}=\frac{10}{3}\cdot3=10[/tex]To find how many cubes fit on the width, we need to divide 4/3 by 1/3, as follows:
[tex]\frac{\frac{4}{3}}{\frac{1}{3}}=\frac{4}{3}\cdot3=4[/tex]To find how many cubes fit on the height, we need to divide 7/3 by 1/3, as follows:
[tex]\frac{\frac{7}{3}}{\frac{1}{3}}=\frac{7}{3}\cdot3=7[/tex]Then, the length has 10 cubes, the width has 4 cubes, and the height has 7 cubes.
The volume of each cube is:
[tex]V=a^3=(\frac{1}{3})^3=\frac{1}{27}ft^3[/tex]The number of cubes that fit in the rectangular prism is: 10x4x7 = 280. Therefore, the volume of the prism is
[tex]280\cdot\frac{1}{27}=\frac{270+10}{27}=\frac{270}{27}+\frac{10}{27}=10\frac{10}{27}ft^3[/tex]Given f(x) f(x) = (- x2 + 7), what is the value of f(4)?
Given the below function;
[tex]f(x)=(-x^2+7)[/tex]If there are 2.54 cm in 1 inch, how long in inches is a meter stick?
To solve the exercise, we can use the rule of three:
Since we know that there are 100 centimeters in a meter, we have:
[tex]\begin{gathered} 2.54\operatorname{cm}\rightarrow1\text{ in} \\ 100\operatorname{cm}\rightarrow x\text{ in} \end{gathered}[/tex][tex]\begin{gathered} x=\frac{100\operatorname{cm}\cdot1in}{2.54\operatorname{cm}} \\ x=\frac{10in\cdot1}{2.54} \\ x=\frac{10in}{2.54} \\ x=39.37in \end{gathered}[/tex]Therefore, there are 39.37 inches in a meter stick.
Given right triangle ABC with altitude BD drawn to hypotenuse AC. If AB = 6 and AD = 2, what is the length of AC? (Note: the figure is not drawn to scale.) B 6 2 D Answer: Submit Answer
The first step is to make a sketch of the triangle
The altitude (h= BD) of the triangle divides it into two similar right triangles and the hypothenuse, AC, into two line segments n= AD and m= DC.
The relationship between the altitude and the parts of the hypothenuse follows the ratios:
[tex]\frac{n}{h}=\frac{h}{m}[/tex]So, the first step is to determine the altitude of the triangle. To do so, you have to work with ΔABD, "h" is one of the sides of the triangle. Using the Pythagorean theorem you can determine the measure of the missing side:
[tex]a^2+b^2=c^2[/tex]Write the expression for the missing side:
[tex]\begin{gathered} b^2=c^2-a^2 \\ \sqrt[]{b^2}=\sqrt[]{c^2-a^2} \\ b=\sqrt[]{c^2-a^2} \end{gathered}[/tex]Replace c=6 and a=2
[tex]\begin{gathered} h=\sqrt[]{6^2-2^2} \\ h=\sqrt[]{36-4} \\ h=\sqrt[]{32} \\ h=4\sqrt[]{2} \end{gathered}[/tex]Now that we have determined the value of the altitude, we can calculate the value of m
[tex]\frac{n}{h}=\frac{h}{m}[/tex]Write the expression for m:
-Multiply both sides by m to take it from the denominators place:
[tex]\begin{gathered} m\cdot\frac{n}{h}=m\cdot\frac{h}{m} \\ m\cdot\frac{n}{h}=h \end{gathered}[/tex]-Multiply both sides of the equal sign by the reciprocal of n/h
[tex]\begin{gathered} m(\frac{n}{h}\cdot\frac{h}{n})=h\cdot\frac{h}{n} \\ m=\frac{h\cdot h}{n} \\ m=\frac{h^2}{n} \end{gathered}[/tex]Replace the expression with h=4√2 and n=2 and calculate the value of m
[tex]\begin{gathered} m=\frac{h^2}{n} \\ m=\frac{(4\sqrt[]{2})^2}{2} \\ m=\frac{32}{2} \\ m=16 \end{gathered}[/tex]So DC=m= 16cm and AD=n= 2cm, now you can determine the measure of the hypothenuse:
[tex]\begin{gathered} AC=AD+DC \\ AC=2+16 \\ AC=18 \end{gathered}[/tex]The hypothenuse is AC=18cm
I need help finding which two could be differences of perfect cubes
Solution:
The differences of perfect cube is expressed in the form:
[tex](a)^3-(b)^3[/tex]From the given options, we have the difference of perfect cubes to be
[tex]\begin{gathered} 216a^6-27y^3\Rightarrow\left(6a^2\right)^3-\left(3y\right)^3 \\ \\ 8a^{15}-27\Rightarrow(2a^5)^3-(3)^3 \end{gathered}[/tex]Hence, the correct options are
Find the solution of the system of equations. 5x + 10y = -5 -5x - y = 32
5x + 10y = -5 ------------------------------(1)
-5x - y = 32 -------------------------------(2)
Add equation(1) and equation (2)
9y = 27
Divide both-side of the equation by 9
y = 3
Substitute y = 3 into equation (1) and solve for x
5x + 10(3) = -5
5x + 30 = -5
substract 30 from both-side of the equation
5x = -5 - 30
5x = -35
Divide both-side of the equation by 5
x = -7
The solution is (-7, 3)