The simplified function of [tex]f(x) = \frac 14(\sqrt[3]{108^x})[/tex] is [tex]f(x) = \frac {3^x}4(\sqrt[3]{4})^x[/tex]
How to simplify the base function?The function is given as:
[tex]f(x) = \frac 14(\sqrt[3]{108^x})[/tex]
Expand 108
[tex]f(x) = \frac 14(\sqrt[3]{27 * 4})^x[/tex]
Take the cube root of 27
[tex]f(x) = \frac 14 * 3^x(\sqrt[3]{4})^x[/tex]
Evaluate the product
[tex]f(x) = \frac {3^x}4(\sqrt[3]{4})^x[/tex]
Hence, the simplified function of [tex]f(x) = \frac 14(\sqrt[3]{108^x})[/tex] is [tex]f(x) = \frac {3^x}4(\sqrt[3]{4})^x[/tex]
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In △ABC, we are told that c=11, ∡C=75∘, and ∡B=29∘. Solve for a and b.
The value of a and b are 5.5 and 11.0.
How to find side of a triangle?
The sides a and b of the triangle can be found as follows:
∠B = 29°
c = 11
∠C = 75°
∠A = 180 - 75 - 29 = 76°
Using sine ratio,
b / sin 29° = 11 / sin 75
b sin 75 = 11 sin 29
b = 11 sin 29 / sin 75
b = 5.33290582271 / 0.96592582628
b = 5.51816958277
b = 5.5
Let's find a,
a / sin 76 = 11 / sin 75
a sin 75 = 11 sin 76
a = 11 sin 76 / sin 75
a = 10.673252989 / 0.96592582628
a = 11.0466922042
a = 11.0
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Write 4.714 as a mixed number
Answer:
47/14 is a mixed number i think
This circle is centered at the origin, and the length of its radius is 10. What is
the circle's equation?
A. x+y=10
B. x¹° + y¹° = 100
C. X²+ y² =10
D. X² +y²=100
Answer:
D
Step-by-step explanation:
If you substitute the radius, 10, and the center of the circle, the origin (also known as (0,0) into the equation of a circle, you should get: [tex](x - 0)^{2} + (y - 0)^{2} = 10 ^{2} [/tex]
Multiplying this out should leave you with D as your answer.
Which graph represents this inequality? 3x + y ≤ 1
Answer:
Step-by-step explanation:
If f(x) = 3x + 2, what is f(5)?
Answer:
If a numbed substitutes x in f (x), you must substitute that number in the function for x.
Step-by-step explanation:
f(5) = 3(5) + 2
f(5 )= 17
You have been saving $25 a week to make the trip; you now have $200. This money will be budgeted to include gas, food, and entertainment. Write an equation to model your savings plan and then graph it. What would be the domain and range in this situation?
Suppose that a forest on a flat terrain has perimeter 200 kilometers, but there is no information on the shape of the forest. What shape will give the largest possible area for the forest? what is this area?
The largest possible area of the forest is 2500 km^2
How to determine the largest area?The perimeter is given as:
P = 200 km
The shape that will give the largest possible area for the forest is a square, and the area is calculated as:
A = (P/4)^2
This gives
A = (200/4)^2
Evaluate
A = 2500 km^2
Hence, the largest possible area of the forest is 2500 km^2
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2y + 3x = -6 in slop intercept form
Answer:
[tex]y=-\frac{3}{2}x-3[/tex]
Step-by-step explanation:
The slope-intercept form is written as y= mx +c, where m is the slope and c is the y-intercept.
For the equation to be in the slope-intercept form, first leave the y term on the left-hand side of the equation and bring the rest to the right-hand side.
2y= -3x -6
Ensure that the coefficient of y is 1. In this question, we can divide both sides of the equation by 2 to achieve the slope-intercept form.
[tex]y=\frac{-3x}{2}-\frac{6}{2}[/tex]
Simplify:
[tex]\bf{y=-\frac{3}{2}x-3 }[/tex]
Additional:
For more questions on slope-intercept form, do check out the following!
https://brainly.com/question/27497166Which of the following results in the difference of two squares?
The difference of two squares expression is (c) (5x + 3y)(5x - 3y)
How to determine the difference of two squares?The difference of two squares expression is represented as:
a^2 - b^2 = (a + b)(a - b)
Using the above as a guide, we have the following expression in option (c)
(5x + 3y)(5x - 3y)
Hence, the difference of two squares expression is (c) (5x + 3y)(5x - 3y)
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Note: When solving for k, round to four decimal places.
A country's population in 1994 was 107 million. In 1997 it was 112 million. Estimate the population in 2008 using the exponential growth formula. Round your answer to the nearest million.
Answer: 132 million
Exponential growth formula: [tex]P=A e^{rt}[/tex]
[P is final amount, A is initial amount, r is growth rate, x is time intervals]
To find growth rate:
[tex]112 = 107 e^{r(3)}[/tex]
[tex]3r= ln(112/107)[/tex]
[tex]r= ln(112/107)/3 = 0.01522334561[/tex]
Population in 2008:
[tex]P_{2008}=107 e^{(0.01522334561(14))}[/tex]
[tex]P_{2008}=132.4169522[/tex]
[tex]P_{2008}=132\:million \quad (rounded \:to\:nearest\:million)[/tex]
PLEASE HELP SOON THX SO MUCH
an airplane flies 3300 miles with the wind in 3 hrs 18 minutes. it takes 4 hrs 24 mins to make the return flight against the same wind. find speed of airplane and wind.
Answer:
The speed of aeroplane =875 miles/hr and the speed of wind =125 miles/hr.
Step-by-step explanation:
Let’s assume the speed of the plane =x miles/hr
And let the speed of wind =y miles/hr
So, the speed of the aeroplane in the direction of wind =(x+y) miles/hr
Speed of the aeroplane in the opposite direction of wind =(x–y) miles/hr
We know that, distance=speed×time
Then according to the given conditions, we have
3300=(x+y)×(3+(18/60))
here convert 18mins to hour by dividing by 60 and add to 3hrs.
⇒x+y= 3300/3.3
⇒x+y=1000 … (i)
And,
3300=(x-y)×(4+(24/60))
here convert 24mins to hour by dividing by 60 and add to 4hrs.
⇒x-y= 3300/4.4
⇒x-y=750 … (ii)
Adding equation (i) and (ii), we get
2x=1750
⇒x= 1750/2
=875
Substituting the value of x in equation (ii), we get
875−y=750
⇒y=875-750
⇒y=125
Therefore, the speed of aeroplane =875 miles/hr and the speed of wind =125 miles/hr.
The shaded region under a Normal distribution with mean 100 and standard deviation 5 is shown. Which of the following is the best choice that corresponds to the shaded region? Select one. mean The shaded region under a Normal distribution with mean 100 and standard deviation 5 is shown . Which of the following is the best choice that corresponds to the shaded region ? Select one .
The shaded region shown in the graph of the normal distribution corresponds to P(x < 92.5).
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
The z score shows by how many standard deviations the raw score is above or below the mean. It is given by:
z = (raw score - mean)/standard deviation
The diagram represents z < 1.5 mean = 100, standard dev = 5 hence:
-1.5 = (x - 100)/5
x = 92.5
The shaded region shown in the graph of the normal distribution corresponds to P(x < 92.5).
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QUESTION IS DOWN BELOW 15 POINTS EACH
Answer:
See below
Step-by-step explanation:
density = mass/volume
re-arrange to density * volume = mass
20 * 64 = 1280 gm
Density = mass/volume
re-arrange to volume = mass / density
560 / 14 = 40 cm^3
helppppppp!!! pleaseeeeeeeee
Answer:
62,400 J
Step-by-step explanation:
As the problem statement tells you, the work done to lift the mass is the product of the force applied and the distance the mass is lifted. Here, the force decreases as the height of the mass increases.
SetupFor x meters of exposed cable, the force required to lift it is 3x newtons. The force required to lift the object is 1500 newtons, so the total force required to lift the masses an incremental distance (dx) is (3x +1500) newtons.
The work done lifting the mass that distance is the product of force and distance. The total work done is the integral of the incremental work over the total of distance lifted:
[tex]\displaystyle W = \int_0^{40}(3x+1500)\,dx[/tex]
SolutionThe value of the integral is ...
[tex]\displaystyle W=\left.\left(\dfrac{3}{2}x^2+1500x\right)\right|_{x=0}^{x=40}=40\left(\dfrac{1}{2}(3\cdot40)+1500\right)=62\,400\qquad\text{joules}[/tex]
The work done to lift the cable and weight is 62,400 joules.
__
Additional comment
One reason for writing the integral evaluation this way is to demonstrate that the total of work done is that required to lift the weight (1500 N, 40 m) and that required to lift the center of mass of the cable (120 N, 20 m).
This will generally be the case for "disappearing mass" or "increasing distance" problems. Pumping from a pool is a similar kind of problem with a similar kind of solution.
Effectively, you're doing work on the center of mass of the system, moving it from its initial position to the top of the building.
Could someone help me out?
The values of the functions are f(4) = 47, f(k) = 2(k)^2 + 4(k) - 1 and f(x + h) = 2x^2 + 4hx + 2h^2 + 4x + 4h - 4
How to evaluate the expression?The function is given as:
f(x) = 2x^2 + 4x - 1
To calculate f(4), we have:
f(4) = 2(4)^2 + 4(4) - 1
Evaluate the expression
f(4) = 47
To calculate f(k), we have:
f(k) = 2(k)^2 + 4(k) - 1
To calculate f(x + h), we have:
f(x + h) = 2(x + h)^2 + 4(x + h) - 1
Expand
f(x + h) = 2(x^2 + 2hx + h^2) + 4x + 4h - 4
Expand
f(x + h) = 2x^2 + 4hx + 2h^2 + 4x + 4h - 4
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angle sum theory find Y
25
Step-by-step explanation:The angle sum theory says that the sum of all the interior angles in a triangle is 180 degrees.
Finding X
To solve for y, we must first find x. This way we know 2 of the interior angles. Luckily, angle x is a part of a linear pair.
Linear Pairs are 2 adjacent angles that create a straight line together. This means that the sum to 180 degrees.Angle x and the angle with a measurement of 115 form a linear pair. Thus, we can create an equation to find x.
115 + x = 180By subtracting 115 from both sides we know that x = 65.
Solving for Y
Now that we know x, we can find y. We know that one of the interior angles is 65 and that the other is 90 degrees. The square marking the bottom angles in the middle show that they are right angles.
Right angles are usuaslly denoted with a square drawn in the angle and have a measurement of 90 degrees.Lastly, we can create a formula to find y with the angle sum theory.
90 + 65 + y = 180Combine like terms
155 + y = 180Subtract 155 from both sides
y = 25This means that the angle y is 25 degrees.
A rectangular piece of metal is 30 in longer than it is wide. Squares with sides 6 in long are cut from the four corners and the flaps are folded upward to form an open box. If the volume of the box is 594 inch to the 3 in, what were the original dimensions of the piece of metal?
The original dimensions of the metal sheet are 22 inches by 52 inch
What is volume?The volume is defined as the space occupied by any object in three dimensions. For a rectangular box, the volume will be calculated by multiplying the length, width, and height of the object.
If the Width of the metal = x
The length is 30 inches longer than its width= x+30
If Squares with sides 6 in long are cut from the four corners and the flaps are folded.
The length of the pre-folded box =x+30-12=(x+18)inch
The width of the pre-folded box = (x-6-6)inch = (x-12) inch
The height of the flap= 6 inch
The volume of the box = 2400-inch cubed
Volume of a Cuboid = LWH
6(x-12)(x+18)=2400
6x²+36x-1296=2400
6x²+36x-1296-2400=0
6x²+36x-3696=0
6(x-22)(28+x)=0
x-22=0 or x+28=0
Therefore the original dimensions of the metal sheet are 22 inches by 52 inch
The complete question with correct data is given below:-
A rectangular piece of metal is 30 in longer than it is wide. Squares with sides 6 in long are cut from the four corners and the flaps are folded upward to form an open box. If the volume of the box is 2400 incubatedwhat were the original dimensions of the piece of metal?
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Find the area of the following figure ( i don't know how to get that answer plsss, help)
The area of the figure is 11. 27 cm²
How to determine the areaThe figure given is a parallelogram
The formula for the area of a parallelogram is given thus;
Area = base × height
From the figure, we have that
base = 2. 3 cm
height = 4. 9 cm
Substitute into the formula
Area = 2. 3 × 4. 9
Area = 11. 27 cm²
Thus, the area of the figure is 11. 27 cm²
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Math interest, I, P, R,T
I will give more points to you once questions are completed.
The interest based on the information given is $2564.
How to calculate the interest?From the information given, the following can be deduced:
Principal = $36900
Time = 178 days
Rate = 14.25%
The simple interest will be:
= (PRT)/100
= (36900 × 14.25 × 178/365)/100
= $2564
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The diagonal of a rhombus is 24m and 10m. Find the perimeter.
Answer:
52m
Step-by-step explanation:
rhombus
its like a slanted square
all its 4 sides are equal like a square
both pairs of opposite sides are parallel
when the diagonals criss cross they make 4 triangles inside
the sides of the rhombus become the hypotenuse of a triangles inside
if the diagonal is 24 and 10 then
its radius is 12 and 5
the sides of the rhombus become the hypotenuse of a triangle that have the sides 12 and 5
hypotenuse of a triangle that has the sides 12 and 5 is
13
(its a special triangle: its a 5, 12, 13 right triangle)
13 times 4 is 52
what is the product? (7x^2)(2x^3+6)(x^2-4x=9)
Answer:
14x^7-56x^6=9
Step-by-step explanation:
distribute
(7x^2)(2x^3 +6)(x^2 -4 * 9=x)
(2x^3+6)(x^2 -4 * 9=x)=2x^5-8x^4=9
(7x^2)(2x^5-8x^4=9)
14x^7-56x^6=9
S and Tare mutually
exclusive events. Find P(sort)
P(S) = 12% P(T) =27%
The value of the probability P(S or T) is 39%
How to determine the probability?The probability values are given as:
P(S) = 12% P(T) =27%
For mutually exclusive events,
P(S or T) = P(S) + P(T)
This gives
P(S or T) = 12% + 27%
Evaluate the sum
P(S or T) = 39%
Hence, the value of the probability P(S or T) is 39%
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Anderson's Solar Systems manufactures solar panels, and their business is growing. Last year,
they hired 90 new employees. This year, they hired 20% more new employees than they hired las
year. How many new employees were hired this year?
The triangles below are similar. Find x.
Answer: 105
Step-by-step explanation:
Corresponding sides of similar triangles are proportional, so
[tex]\frac{x}{14}=\frac{97.5}{13}\\\\=\frac{x}{14}=7.5\\\\x=\boxed{105}[/tex]
Okay, I am getting familiar with quadratic equations by I'm not sure of X=m+/- n... Help!
The value of m and n to show the solution of the equation x² + 6x - 5 = 0 is m = -3 and n = ± √14
What is an equation?An equation is an expression that shows the relationship between two or more variables and numbers.
Given the equation:
x² + 6x - 5 = 0
x² + 6x = 5
x² + 6x + 9 = 5 + 9
(x + 3)² = 14
x + 3 = ±√14
x = -3 ± √14
The value of m and n to show the solution of the equation x² + 6x - 5 = 0 is m = -3 and n = ± √14
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Decide if the function is an exponential function. If it is, state the initial value and the base. (5 points) y = - 6.1 ⋅ 3x
The function is an exponential function; the initial value is -6.1 and the base is 3
How to determine the function type?The equation of the function is given as:
y = -6.1 *(3^x)
An exponential function is represented as:
y = ab^x
Where:
Initial value = a
base = b
By comparison, we have
a = -6.1 and b = 3
Hence, the function is an exponential function; the initial value is -6.1 and the base is 3
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a square pyramid has side length 9 in. and volume = 270 in^3. find the height.
If the square pyramid has side length 9 inches and volume of 270 inches cube then by putting the values within the formula of volume height are going to be 10 inches.
Given Length and volume of a square pyramid are 9 inches and 270 inches cube. Square pyramid is the pyramid having square base. It is a right square pyramid.
We know that the amount of square pyramid is 1/3 *[tex]l^{2} *h[/tex]
Put the values in formula to induce the values oh h.
Volume=1/3*[tex]9^{2}*h[/tex]
270=1/3* 81*h
h=(270*3)/81
h=810/81
h=10
Hence the peak of the square pyramid having length 9 inches and volume of 270 inches cube is 10 inches.
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How can I solve this?? Please help
Answer:yes
15x times 2=30x
15x times 3= 45x
3y tines 2 =6y
3y times 3=9y
Step-by-step explanation:
Sound waves can be ranked by their intensity, I, given in this formula, where r is the distance from the source of a sound with a power output of P=4 πIr^2 Which equation correctly rewrites the formula to solve for I?
Answer:
See below
Step-by-step explanation:
P = 4pir^2 l divide both sides by 4 pi r^2
l = P /(4pir^2)
what's the numerator for the following rational expression? 6/t+7/t=?/t
The numerator for the following rational expression is 13
How to determine the numeration?The rational expression is given as:
6/t + 7/t
Take the LCM of the denominator;
The LCM of t and t is t
So, we have:
6/t + 7/t = (6 + 7)/t
Evaluate the sum
6/t + 7/t = 13/t
13 is the numerator, while t is the denominator
Hence, the numerator for the following rational expression is 13
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