Answer:
[tex]x=\dfrac{\ln \left(\frac{21}{25}\right)}{\ln 6125}[/tex]
Step-by-step explanation:
Given equation:
[tex]2 \cdot 5^{3x+2} = 6 \cdot 7^{1-2x}[/tex]
Divide both sides of the equation by 2:
[tex]\implies 5^{3x+2} = 3 \cdot 7^{1-2x}[/tex]
Take natural logs of both sides:
[tex]\implies \ln \left(5^{3x+2}\right) = \ln \left(3 \cdot 7^{1-2x}\right)[/tex]
[tex]\textsf{Apply the product law:} \quad \ln xy=\ln x + \ln y[/tex]
[tex]\implies \ln 5^{3x+2} = \ln 3 + \ln 7^{1-2x}[/tex]
[tex]\textsf{Apply the power law:} \quad \ln x^n=n \ln x[/tex]
[tex]\implies (3x+2)\ln 5 = \ln 3 + (1-2x)\ln7[/tex]
Expand:
[tex]\implies 3x\ln 5+2\ln 5=\ln 3+\ln7-2x\ln7[/tex]
Collect the terms in x on the left side and the constants on the right side of the equation:
[tex]\implies 3x\ln 5+2x\ln7=\ln 3+\ln7-2\ln 5[/tex]
Factor out the common term x on the left side:
[tex]\implies x(3\ln 5+2\ln7)=\ln 3+\ln7-2\ln 5[/tex]
[tex]\textsf{Apply the power law:} \quad n \ln x=\ln x^n[/tex]
[tex]\implies x(\ln 5^3+\ln7^2)=\ln 3+\ln7-\ln 5^2[/tex]
Simplify:
[tex]\implies x(\ln 125+\ln 49)=\ln 3+\ln7-\ln 25[/tex]
[tex]\textsf{Apply the product law:} \quad \ln x + \ln y=\ln xy[/tex]
[tex]\implies x(\ln (125 \cdot 49))=\ln (3\cdot 7)-\ln 25[/tex]
[tex]\implies x(\ln 6125)=\ln 21-\ln 25[/tex]
[tex]\textsf{Apply the quotient law:} \quad \ln x - \ln y=\ln \left(\dfrac{x}{y}\right)[/tex]
[tex]\implies x(\ln 6125)=\ln \left(\dfrac{21}{25}\right)[/tex]
Divide both sides of the equation by ln 6125:
[tex]\implies x=\dfrac{\ln \left(\frac{21}{25}\right)}{\ln 6125}[/tex]
A classmate claims that ( f ⋅ g)' = f ' ⋅ g' for any functions f and g. Show an example that proves your classmate wrong.
This is not true for all functions. To prove this, an example is given below. Let f(x) = x and g(x) = x².
What is function?A function is a relation that associates each element of a set, called the domain, to exactly one element of another set, called the range. It is a type of correspondence between two sets, where each element of one set is associated with exactly one element of another set.
The first derivative of f(x) is f ' (x) = 1 and the first derivative of g(x) is g ' (x) = 2x.
Therefore, ( f ⋅ g )' = ( x ⋅ x2 )'
= 1 ⋅ 2x + x ⋅ 2
= 2x + 2 ≠ 1 ⋅ 2x + 2x
= 4x
= f ' ⋅ g '
This example proves that the statement (f⋅g)' = f'⋅g' is incorrect. It is important to note that this does not mean that the statement is not true for all functions. It simply means that for the particular example given, the statement is not correct.
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Let c stand for the number of cousins sawyer has. Which bar model represents the problem?
The bar model that represents the number of cousins that Sawyer has, given the cousins Hazel has is the fourth bar model.
How to find the bar model ?The letter " c " is meant to represent the number of cousins that Sawyer has, in relation to those that Hazel has. This means that bar models A and C are out because they represent the cousins as B.
The question shows that Hazel has twice the number of cousins that Sawyer has. This means that if Sawyer's cousins are represented as " c " then Hazel would be 2 c. This is shown in the last bar model so this is the correct bar model.
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First part is:
Hazel has 8 cousins. Hazel has 2 times as many cousins as Sawyer.
How many cousins does Sawyer have?
Lily has 4 cups of apples, 5-½ cups of watermelon, and cups of
grapes. She needs 2½ times of each fruit to make her fruit bowl. How
much of each fruit does she need? Do not round. Write your answer as
a whole number or as a mixed number. Use "/" as a fraction bar. Do
not label your answers.
a. Apples
b. Watermelons
c. Grapes
a. Apples: Lily needs 10 cups of apples for her fruit bowl.
b. Watermelons: Lily needs 13-3/4 cups of watermelons for her fruit bowl.
c. Grapes: The amount of grapes that Lily has is not given
What is arithmetic sequence?
An arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a fixed constant number, called the common difference, to the preceding term.
To make her fruit bowl, Lily needs 2½ times the amount of each fruit she currently has.
a. Apples:
2½ × 4 cups = 10 cups
Therefore, Lily needs 10 cups of apples for her fruit bowl.
b. Watermelons:
2½ × 5-½ cups = (2 × 5 + 1/2) × 5/2 cups = 13-3/4 cups
Therefore, Lily needs 13-3/4 cups of watermelons for her fruit bowl.
c. Grapes:
The amount of grapes that Lily has is not given, so we cannot calculate how much she needs for her fruit bowl.
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A certain car was purchased for $28,500. The car
has a resale value of $17,500 after a useful life of 6
years. What is the annual depreciation expense for
this car?
Annual depreciation expense = $ [ ? ]
Round to the nearest hundredth.
The annual depreciation expense for this car is $1,833.33.
What is the depreciation formula?Simply deduct the vehicle's current fair market value from the purchase price, less any applicable sales tax or fees, to determine how much value has been lost.
We must determine the total depreciation throughout the vehicle's useful life and divide it by the number of years to determine the annual depreciation expense:
Total depreciation = purchase price - resale value
= $28,500 - $17,500
= $11,000
Annual depreciation expense = total depreciation / useful life = $11,000 / 6 = $1,833.33 (rounded to the nearest hundredth)
Therefore, the annual depreciation expense for this car is $1,833.33.
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0.9t
t² + 64
Find the time
The concentration of a drug t hours after being injected is given by C(t) =
when the concentration is at a maximum. Give your answer accurate to at least 2 decimal places.
hours.
Calculator
Submit Question
Let the Universal set be the letters a through j: U = {a, b, ..., i, j}.
The elements of the sets A ∩ (B ∪ C) is {b, c, i}.
How to find the element of a set?A universal set (U) is a set which has elements of all the related sets, without any repetition of elements.
Therefore,
U = {a, b, ….. i, j}
A = {b, c, i, j}
B = {a, c, g, i}
C = {a, b, d, i}
Therefore, let's find the elements A ∩ (B ∪ C).
Hence,
(B ∪ C) = {a, b, c, d, i, g}
A = {b, c, i, j}
Finally,
A ∩ (B ∪ C). = {b, c, i}
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Please see the attached
The amount in the account at the end of the year is $4998.
What is percentage decrease?Percentage decrease is the measure of the reduction in the value of a quantity over time, expressed as a percentage of its original value. It is calculated by taking the difference between the original value and the new value, dividing it by the original value, and then multiplying the result by 100.
We know that;
Percentage decrease = 26.5%
Then the decrease in dollars = 26.5/100 * 6800
= $1802
The amount at the end of the year = $6800 - $1802
= $4998
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write a quadratic function in standard form that passes through (-4,0) , (5,0) , and (3,14) .
Evaluate the limit
And show your work pleaseee
Answer:0
Step-by-step explanation: -
-7 x -7 = 49
11 x 77 = 77
-7 + 7 = 0
49 + 77 + 28 = 154
154 ÷ 0 = 0
Graph the line with slope 3 passing through the point (5, 3)
A graph of the line with slope 3 passing through the point (5, 3) is shown in the image attached below.
How to determine an equation of this line?In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical expression:
y - y₁ = m(x - x₁)
Where:
x and y represent the data points.m represent the slope.At data point (5, 3) and a slope of 3, a linear equation for this line can be calculated by using the point-slope form as follows:
y - y₁ = m(x - x₁)
y - 3 = 3(x - 5)
y = 3x - 15 + 3
y = 3x - 12
In conclusion, we would use an online graphing calculator to plot the equation y = 3x - 12 as shown in the graph attached below.
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Given the circle below with secants JKL and NML, find the length of JK. Round to the nearest tenth if necessary.
Using the Intersecting secants theorem, the length of JK in the given diagram is 17
Intersecting secants theorem: Calculating the length of of JKFrom the question, we are to determine the length of JK in the given diagram.
The diagram shows a circle with intersecting secants.
From the Intersecting secants theorem which states that "if two secant segments are drawn to a circle from an exterior point, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment".
Then,
We can write that
JK × KL = NM × ML
From the given diagram,
KL = 14
NM = 14
ML = 17
Substituting the values
JK × 14 = 14 × 17
JK = (14 × 17) / 14
JK = 17
Hence, the length of JK is 17
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What are the roots of the quadratic equation x² + 4x + 7 = 0?
A) 2 + i√3 or 2 - i√3
B) x = -2 + i√3 or -2 - i√3
C) x = -7 or 1
D) non of above
Answer:
The roots of the quadratic equation x² + 4x + 7 = 0 can be found using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
In this case, a = 1, b = 4, and c = 7. Substituting these values into the formula, we get:
x = (-4 ± √(4² - 4(1)(7))) / 2(1)
x = (-4 ± √(16 - 28)) / 2
x = (-4 ± √(-12)) / 2
x = (-4 ± 2i√3) / 2
x = -2 ± i√3
Therefore, the roots of the equation x² + 4x + 7 = 0 are: A) 2 + i√3 or 2 - i√3.
So, the correct answer is A
4. A rectangular prism has height of 2¾ cm, the length is 1¼ cm and the height 4 cm, what is the volume?
Answer:
13.75 cm³
Step-by-step explanation:
In order to find the area of a rectangular prism you only need to multiply the width height and length together:
2.75 * 1.25 = 3.4375
3.4375 * 4 = 13.75
Hope this helps!
A chess board is made using the ratio of square length to king's height, 3.5 inches to 4.5 inches. If a chess board is made with a king's height of 2.25 inches, what is the length of the square?
The length of the square is 1.75 inches, while the king's height is 2.25 inches.
Define length.
The length of an object or the extent of a one-dimensional object along its longest dimension is measured physically. It is a basic idea in physics and geometry and is frequently expressed in terms of meters (m), centimetres (cm), feet (ft), or inches (in).
According to the information provided, the square length to king's height ratio is 3.5 to 4.5 inches. Let's use this ratio to determine the square's length when the king is 2.25 inches tall.
We may make a proportion to get the square's length:
(square length) / (king's height) = 3.5/4.5
king's height = 2.25
Let's substitute the given values:
(square length) / 2.25 = 3.5/4.5
To solve for the length of the square, we can cross-multiply:
(square length) = (2.25) x (3.5/4.5)
(square length) = 1.75
Therefore, the length of the square is 1.75 inches, while the king's height is 2.25 inches.
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What is the length of GH?
Answer: 52-9 = 43 units
Step-by-step explanation:
6) Given the points A(-4, 5) and B(6, 0), find
bolthe coordinates of the point P on the line
segment AB that partitions AB into the
ratio 2:3. Plot P along with segment AB.
Answer: (0, 3)
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A report says that the average amount of time a 10-year-old American child spends playing outdoors per day is between 20.92 and 24.48 minutes. What is the margin of error in this report?
5. The class finishes seeing the Small
Animals at 1:30. They will get back at
school in 30 minutes.
What time will it be then? Choose a
way to show the time.
What is the solution set for |z+4| > 15?
O 11 < z < 19
O -19 < z < 11
O z < -19 or z > 11
O z < 19 or z > 11
Answer:
C. z < -19 or z > 11
Step-by-step explanation:
|z+4| > 15
z+4<-15 or z+4<15
z < -19 or z > 11
a cuboid with square base has a volume of (147p-245q) cubic centimeters. The length of the square bade is 7 meters. Find an expression for the height of the cuboid.
An expression for the height of the cuboid is 3p - 245q/49 meters.
How to calculate the volume of cuboid with a square base?In Mathematics and Geometry, the volume of a cuboid with a square base can be calculated by using this following formula:
Volume of a cuboid = L² × H
Where:
L represents the base area of a cuboid with a square base.H represents the height of a cuboid with a square base.By substituting the given parameters into the formula for the volume of a cuboid cuboid with a square base, we have the following;
(147p - 245q) = 7² × H
(147p - 245q) = 49 × H
Height, H = (147p - 245q)/49
Height, H = 3p - 245q/49 meters.
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A trapezoid is 64 m2. The height is 12 cm and one base is 4 cm. What is the length of the missing base?
The length of the missing base is approximately 6.67 cm.
How to calculate the length of trapezoidBy applying the formula for the area of a trapezoid, we will be able to solve for the missing base.
Recall that the formula for the area of a trapezoid is:
A = [tex]\frac{1}{2} * h * (b_{1} + b_{2})[/tex]
where
A = area
h = height
b₁ and b₂ are the lengths of the parallel bases.
From the question, we are given:
A = 64 m²
h = 12 cm
b₁ = 4 cm.
Plug these values into the formula above to solve the missing base
64 = [tex]\frac{1}{2} * 12 * (4+ b_{2})[/tex]
Multiplying both sides by 2:
128 = 12(4 + b₂)
Simplifying:
128 = 48 + 12b₂
80 = 12b₂
b₂ = 80/12 = 20/3
b₂ = 6.67
Therefore, the length of the missing base is approximately 6.67 cm.
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-10-9-8
Which inequality is graphed on the number line?
OA) x>-7
OB) x < -7
C) x ≤ -7
D) xz -7
Answer:
a) x > -7
Step-by-step explanation:
an open circle (blank) means the greater than or less than symbols (< or >) and a close circle (filled in) means ≤ or ≥. the lining to your right looks more bold than the lining to your left so it's an; open circle, on -7, with the lining bold to your left.
in conclusion, x (the open circle) > -7
please help me solve this
From the calculations, only question 2 shows that f(x) is the inverse of function g(x) as the rest shows the parent functions are not inverse of g(x)
What are the inverse of the functionIn the given question, we are to determine if the parent function f(x) is an inverse to g(x).
1. f(x) = 2x - 9
To find the inverse function, swap x and y, and solve the resulting equation for x.
If the initial function is not one-to-one, then there will be more than one inverse.
Let's switch the variables: y = 2x - 9 to x= 2y - 9.
We can proceed to solve the equation as;
x =2y - 9 for y.
y= x + 9 / 2
Therefore, f(x) is not an inverse of g(x)
2. f(x) = x - 3 / 4
To find the inverse function, swap x and y, and solve the resulting equation for x.
If the initial function is not one-to-one, then there will be more than one inverse.
Let's switch the variables:
y = x / 4 - 3/4 to x = y / 4 - 3/4
we can solve the equation for y to give us;
y = 4x + 3
This shows that f(x) is inverse of g(x)
4. f(x) = 7x^3/2 - 4
To find the inverse function, swap x and y and solve the resulting equation for x
If the initial function is not one-to-one, then there will be more than one inverse.
So, swap the variables:
y = 7x^3/2 - 4 will become x = 7y^3/2 - 4
Solving for y;
y = [(∛7(x + 4))^2/3] / 7
This shows f(x) is not the inverse of g(x)
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Pranav and Jill are selling pies for a school findraiser, Customers can buy apple pies and blackborry pies. Pranav sold S apple ples and 4 blackberry pies for a total of 8140, Ill sold o apple pies and 4 blackberry pies for a total of $152. What is the cost each of one apple tie and one blackberry pie?
Answer:
Let the cost of one apple pie be A and the cost of one blackberry pie be B.
From the first statement, we can write the equation:
SA + 4B = 8140 ----(1)
From the second statement, we can write the equation:
4*B = 152
Solving for B:
B = 38
Substituting B in equation (1) and solving for A:
SA + 438 = 8140
S*A = 8008
A = 8008/S
We do not have enough information to find the value of S or A individually, but we can find the relationship between them:
A/B = (8008/S)/38 = 211/S
Therefore, the cost of one apple pie is 211 times the cost of one blackberry pie.
12. Math on the Spot Libby puts a mat of width m and
a frame of width faround an 8-inch by 10-inch picture.
Find an expression for the perimeter of the entire frame.
Answer:
The perimeter of the entire frame is equal to the sum of the perimeters of the mat and the picture. The expression for the perimeter of the mat is (2m + 10) + (2m + 8) = 4m + 18, and the expression for the perimeter of the picture is 2(8) + 2(10) = 36. Therefore, the expression for the perimeter of the entire frame is (4m + 18) + 36 = 4m + 54.
What is the value of x in the figure? X=____
that L looking shape is a right angle, which is 90 degrees. and the question is saying that the part they cut off is 56 degrees. to find x you will need to subtract 90 from 56 to get 34.
The answer is 34 :)
Select the correct answer.
Which function defines (g - f)(x)?
Answer:
Option A
Concept Applied:
(g-f)(x) = g(x)-f(x)
Step-by-step explanation:
Substitute and solve
Need help will give brainliest and 5 stars! :)
Answer:
2.362
Step-by-step explanation:
Use calculator: 2.361727836
Round: 2.362
I DONT GET THIS HELP ME PLEASE. there are 6 spots for every what eyes. there are what sports for 1 eye
The ratio of spots to eyes can be described as follows;
There are 6 spots for every 2 eyes.
There are 3 spots for every 1 eye.
What is a ratio?In Mathematics and Geometry, a ratio can be defined as a mathematical expression that's used to denote the proportion of two (2) or more quantities with respect to one another and the total quantities.
Based on the information provided above, we have the following ratio parameters:
The total number of spots = 6 spots.
The total number of eyes = 2 eyes.
Ratio = 6/2 = 3 spots.
In this context, we can reasonably infer and logically deduce that there are 6 spots for every two (2) eyes and three (3) spots for every one (1) eye.
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Complete Question:
Use the following image to complete the sentences describing the ratio of spots to eyes.
There are 6 spots for every ____ eyes.
There are _____ spots for every 1 eye.
Help me! I need to find the surface area if a cylinder!
The surface area of the cylinder is 1851.97 square meters
Finding the surface area of the cylinderFrom the question, we have the following parameters that can be used in our computation:
Radius, r = 13.1 meters
Height, h = 9.4 meters
Using the above as a guide, we have the following:
Surface area = 2πr(r + h)
Substitute the known values in the above equation, so, we have the following representation
Surface area = 2π * 13.1 * (13.1 + 9.4)
Evaluate
Surface area = 1851.97
Hence, the surface area is 1851.97 square meters
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