Assume that the flask shown in the diagram can be modeled as a combination of a sphere and a cylinder. Based on this assumption, the volume of the flask is ____ cubic inches. If both the sphere and the cylinder are dilated by a scale factor of 2, the resulting volume would be _____ times the original volume.
options for the first blank are: 20.22, 35.08, 50.07, or 100.11
options for the second blank are: 2, 4, 6 or 8
The total volume of the flask will be 50.06 [tex]\rm inches ^3[/tex] and if both the sphere and the cylinder are dilated by a scale factor of 2, the resulting volume would be '8' times the original volume.
Given :
Flask can be modeled as a combination of a sphere and a cylinder.
The volume of Sphere is given by the:
[tex]V_s = \dfrac{4}{3}\pi r^3[/tex]
Given - diameter of sphere = 4.5 inches. Therefore, radius is 2.25 inches.
Now, the volume of sphere of radius 2.25 inches will be:
[tex]V_s = \dfrac{4}{3}\times \pi\times (2.25)^3[/tex]
[tex]\rm V_s = 47.71\; inches^3[/tex]
The volume of Cylinder is given by the:
[tex]V_c = \pi r^2h[/tex]
Given - diameter of cylinder = 1 inches then radius is 0.5 inches and height is 3 inches.
Now, the volume of cylinder of radius 0.5 inches and height 3 inches will be:
[tex]V_c = \pi\times (0.5)^2 \times 3[/tex]
[tex]\rm V_c = 2.35\; inches^3[/tex]
Therefore the total volume of the flask will be = 47.71 + 2.35 = 50.06 [tex]\rm inches ^3[/tex].
Now, if both the sphere and the cylinder are dilated by a scale factor of 2 than:
Radius of sphere = [tex]2.25\times 2[/tex] = 4.5 inches
Radius of cylinder = [tex]0.5\times 2[/tex] = 1 inch
Height of cylinder = [tex]3\times 2[/tex] = 6 inches
Now, the volume of sphere when radius is 4.5 inches will be:
[tex]V_s' = \dfrac{4}{3}\times \pi \times (4.5)^3[/tex]
[tex]\rm V_s' = 381.70\; inches ^3[/tex]
And the volume of cylinder when radius is 1 inch and height is 6 inches will be:
[tex]V_c' = \pi \times (1)^2\times 6[/tex]
[tex]\rm V_c'=18.85\;inches^3[/tex]
Therefore the total volume of the flask after dilation by a scale factor of 2 will be = 381.70 + 18.85 = 400.55 [tex]\rm inches ^3[/tex].
Now, divide volume with dilation by theorginal volume of the flask.
[tex]\dfrac{400.55}{50.06}=8[/tex]
Therefore, if both the sphere and the cylinder are dilated by a scale factor of 2, the resulting volume would be '8' times the original volume.
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A line passes through (2, –1) and (8, 4).Write an equation for the line in point-slope form.
Rewrite the equation in standard form using integers.
Hello : let
A(2,-1) B(8,4)
the slope is : (YB - YA)/(XB -XA)
(4+1)/(8-2) = 5/6
Answer: Equation of line in point slope form,
[tex]y + 1 = 5 ( x - 2 )[/tex]
And, Equation of line in standard form,
[tex]5 x - 6 y = 16[/tex]
Step-by-step explanation:
Since, If a line passes through two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] ,
Then the equation of line,
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1} (x-x_1)[/tex]
Here [tex]x_1 = 2[/tex], [tex]y_1=-1[/tex], [tex]x_2=8[/tex] and [tex]y_2=4[/tex]
Thus, the equation of the given line,
[tex]y-(-1)=\frac{4-(-1)}{8-2} (x-2)[/tex]
⇒ [tex]y+1=\frac{4+1}{8-2} (x-2)[/tex]
⇒ [tex]y+1=\frac{5}{6} (x-2)[/tex] -----(1)
⇒ [tex]6(y+1)= 5(x-2)[/tex]
⇒ 6 y + 6 = 5 x - 10
⇒ 6 = 5x - 6y - 10 ( By subtracting by on both sides )
⇒ 6 + 10 = 5x - 6y ( By adding 10 on both sides )
⇒ 16 = 5x - 6y
⇒ 5 x - 6 y = 16 ------(2)
Since, in slope for of a line is, [tex]y-y_1= m (x-x_1)[/tex]
Thus, equation (1) shows the in slope form of the line.
And, standard form of the line is ax + by = c where a, b and c are the integers.
Thus, equation (2) shows the standard form of the given line.
1. is acute. 2. is isosceles. 3. is right. Which two statements contradict each other?
Answer:
the answer is 2 and 3
Step-by-step explanation:
I took the test
Jean has 5 different colors of markers: red, blue, green, orange, and purple. Two colors are used to make a sign. How many different combinations are possible? List them.
A basket contains 11 pieces of fruit: 4 apples, 5 oranges, and 2 bananas. Jonas takes a piece of fruit at random from the basket, and then Beth takes a piece at random. What is the probability that Jonas will get an orange and Beth will get an apple?
The probability that Jonas will get an orange and Beth will get an apple is 20/121.
Explanation:To find the probability that Jonas will get an orange and Beth will get an apple, we need to find the probability of Jonas getting an orange and then multiply it by the probability of Beth getting an apple.
The probability of Jonas getting an orange is the number of oranges in the basket (5) divided by the total number of fruits (11):
P(orange for Jonas) = 5/11
The probability of Beth getting an apple is the number of apples in the basket (4) divided by the total number of fruits (11):
P(apple for Beth) = 4/11
To find the combined probability, we multiply these two probabilities together:
P(orange and apple) = P(orange for Jonas) * P(apple for Beth) = (5/11) * (4/11) = 20/121
The sum of four consecutive whole numbers is 54, what are the four numbers
let n be the first 3 of consecutive even integers. what is the sum of those integers?
Find the total area of the prism in ( _+_√_ )
(25 Points)Enter numbers to write 4.23×10^3 in standard notation.
What are the roots of the function y = 4x2 + 2x – 30? To find the roots of the function, set y = 0. The equation is 0 = 4x2 + 2x – 30.
Factor out the GCF of .
Next, factor the trinomial completely. The equation becomes .
Use the zero product property and set each factor equal to zero and solve. The roots of the function are .
The roots of the equation [tex]y=4x^2+2x-30[/tex] are x = -3 and x = 5/2
Roots of a quadratic equationThe given quadratic equation is:
[tex]y=4x^2+2x-30[/tex]
Set y = 0
[tex]4x^2+2x-30=0[/tex]
Factor the trinomial completely
[tex]4x^2-10x+12x-30=0\\\\2x(2x-5)+6(2x-5)=0\\\\(2x-5)(2x+6)=0[/tex]
Set each factor to zero and solve
2x - 5 = 0
2x = 5
x = 5/2
2x + 6 = 0
2x = -6
x = -6/2
x = -3
The roots of the equation [tex]y=4x^2+2x-30[/tex] are x = -3 and x = 5/2
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A) side-side-side triangle similarity postulate
B) angle-angle triangle similarity postulate
C) angle-side-angle triangle similarity postulate
D) hypotenuse-lag triangle similarity postulate
I don't get it I got a different answer then these
what is the period of the sinusoid given by y=-4sin( [tex] \frac{2π}{3} [/tex] x) ?
Answer:
The answer is 3 for A P E X
Step-by-step explanation:
How could the relationship of the data be classified?
scatter plot with points loosely scattered going down to the right
A positive correlation
A causation
A negative correlation
No correlation
Answer: A negative correlation
Step-by-step explanation:
If the points in the scatter plot scattered going down to the right, it shows that there are inverse relationship between the quantities.
With the increase of one quantity or variable there is decrease in the other quantity or variable.
Therefore, if in the scatter plot with points loosely scattered going down to the right , then the relationship of the data be classified as a negative correlation.
A rectangular garden has length twice as great as its width. A second rectangular garden has the same width as the first garden and length that is 4 meters greater than the length of the first garden. The second garden has area of 70 square meters. What is the width of the two gardens?
(02.02 MC)
Line segment RS is shown on a coordinate grid:
The line segment is rotated 270 degrees counterclockwise about the origin to form R'S'. Which statement describes R'S'?
R'S' is parallel to RS.
R'S' is half the length of RS
. R'S' is twice the length of RS.
R'S' is equal in length to RS.
Use the functions f(x) = 4x − 5 and g(x) = 3x + 9 to complete the function operations listed below.
Part A: Find (f + g)(x). Show your work. (3 points)
Part B: Find (f ⋅ g)(x). Show your work. (3 points)
Part C: Find f[g(x)]. Show your work. (4 points)
(07.05 MC)
An equation is shown below:
4x + 2(x – 3) = 4x + 2x – 11
Part A: Solve the equation and write the number of solutions. Show all the steps. (6 points)
Part B: Name one property you used to solve this equation. (4 points
Answer:
-6 = -11
Distributive Property
Step-by-step explanation:
A.
4x + 2(x – 3) = 4x + 2x – 11
4x + 2x - 6 = 4x + 2x - 11
6x - 6 = 6x - 11
-6 = -11
since -6 is not equal to -11, so there's no solution
B. Distributive property
Use complete sentences to describe the range of the sine function.
traveling at 65 miles per hour how many minutes rounded to the nearest whole number does it takes to drive 125 miles from san digit to malibu
divide total miles by speed
125/65 = 1.923 hours
there are 60 minutes per hour
multiply 1.923*60 = 115.384 minutes
rounded off to nearest whole number = 115 minutes
A rectangle is 42 square feet. what percent of the area of the rectangle is a square with side lengths of 6 feet?
area = L x w
42 = 6 x w
w=42/6 = 7
rectangle is 6 x 7
square would be 6 x 6 = 36 square feet
36/42 = 0.857 = 85.7% ( Round answer if needed)
Two 6-sided dice are rolled. what is the probability the sum of the two numbers on the die will be 4?
Answer:
[tex]\frac{1}{12}[/tex].
Step-by-step explanation:
Given : Two 6-sided dice are rolled.
To find : what is the probability the sum of the two numbers on the die will be 4.
Solution : We have given
Two 6-sided dice are rolled.
Dice have number { 1,2,3,4,5,6} { 1,2,3,4,5,6} .
[tex]Probability =\frac{outcome\ happn}{total\ outcome}[/tex].
sum of the two numbers on the die will be 4.
Case (1) : first dice rolled 3 and second dice rolled 1.
{3,1}
3 +1 = 4 .
Case (2) : first dice rolled 1 and second dice rolled 3 .
{1,3}
1 + 3 = 4 .
Case (3) : first dice rolled 2 and second dice rolled 2.
{2,2}
2 + 2 = 4.
Then there are 3 possible outcomes where the sum of the two dice is equal to 4.
The number of total possible outcomes = 36.
[tex]Probability =\frac{3}{36}[/tex].
[tex]Probability =\frac{1}{12}[/tex].
Probability of getting sum of two dice is [tex]\frac{1}{12}[/tex].
Therefore, [tex]\frac{1}{12}[/tex].
The sum of two rational numbers will always be
Line BC has an equation of a line y = 2x + 3, and line EF has an equation of a line y = negative one over 2 x + 4. These two equations represent
Answer:
Perpendicular lines.
Step-by-step explanation:
We have been given that line BC has an equation of a line [tex]y=2x+3[/tex] and line EF has an equation of a line [tex]y=-\frac{1}{2}x+4[/tex]. We are asked to determine what these both equations represent.
We know that slope of two perpendicular lines is negative reciprocal of each other. This means the product of slope of both lines is equal to [tex]-1[/tex].
Let us find the product of slopes of both lines.
[tex]2\times \frac{-1}{2}[/tex]
Upon cancelling 2 with 2 we will get,
[tex]=-1[/tex]
Therefore, the given two equations represent equations of two perpendicular lines.
Write an equation for the line that is parallel to the given line and that passes through the given point.
y = 1/2 – 8; (–6, –17)
A.) y = 2x – 14
B.) y = 1/5x + 5/2
C.) y = -2x + 14
D.) y = 1/2x – 14
Solve the equation ,and check the solution.
r - 3 r
____ = __
10 13 Check:
To solve the equation, we cross multiply to eliminate the fractions and solve for r. The solution is r = 13. We can check the solution by substituting it back into the original equation.
Explanation:To solve the equation (r - 3) / 10 = r / 13, we can cross multiply to eliminate the fractions. This gives us 13(r - 3) = 10r. Distributing, we get 13r - 39 = 10r. We can then solve for r by subtracting 10r from both sides and adding 39 to both sides. This gives us 3r = 39. Dividing both sides by 3, we find that r = 13.
To check the solution, we substitute r = 13 back into the original equation. The left side becomes (13 - 3) / 10 = 10 / 10 = 1, and the right side becomes 13 / 13 = 1. Since both sides are equal to 1, we can confirm that the solution r = 13 is correct.
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The transformation from f to g represents a __________ stretch. f(x) = Square root of x. and g(x) = 6Square root of x.
The transformation from f to g represents a Vertical stretch.
Step-by-step explanation:We are given a parent function f(x) as:
[tex]f(x)=\sqrt{x}[/tex]
and the transformed function g(x) as:
[tex]g(x)=6\sqrt{x}[/tex]
We know that any function transformation of the type:
f(x) → a f(x)
represents either a vertical stretch stretch or compression depending on the value of a.
If 0<a<1 then the transformation is a vertical compression and if a>1 then the transformation is a vertical stretch.
Here we have a=6>1
Hence, the transformation is a VERTICAL STRETCH.
What is the equation of the line that passes through the point of intersection of the lines y = 2x − 5 and y = −x + 1, and is also parallel to the line y=1/2x+4?
SOMEBODY HELP ME WITH THESE PLEASE! I really need the help like NOW PLS!
Answer:
97.
For finding f(g(x)) we will plugin value of g(x) in place of x in f(x).
99.
Step-by-step explanation:
(Solve for r) 0.5r − 3.8 = 5.66