Answers
Answer:
[tex]x=10^{\frac{4}{3}}=\sqrt[3]{10000}[/tex]
Step-by-step explanation:
Given equation:
[tex]9^{\log_3 (\log x)}= \log x-2\log^2x+4[/tex]
Rewrite 9 as 3²:
[tex]\implies (3^2)^{\log_3 (\log x)}= \log x-2\log^2x+4[/tex]
[tex]\textsf{Apply exponent rule} \quad (a^b)^c=a^{bc}:[/tex]
[tex]\implies 3^{2\log_3 (\log x)}= \log x-2\log^2x+4[/tex]
[tex]\textsf{Apply the log power law}: \quad \log_ax^n=n\log_ax[/tex]
[tex]\implies 3^{\log_3 (\log x)^2}= \log x-2\log^2x+4[/tex]
[tex]\textsf{Apply log law}: \quad a^{\log_ab}=b[/tex]
[tex]\implies (\log x)^2= \log x-2\log^2x+4[/tex]
Simplify:
[tex]\implies\log ^2 x= \log x-2\log^2x+4[/tex]
[tex]\implies 3\log ^2 x- \log x-4=0[/tex]
Let log(x) = u:
[tex]\implies 3u^2-u-4=0[/tex]
Factor:
[tex]\implies 3u^2+3u-4u-4=0[/tex]
[tex]\implies 3u(u+1)-4(u+1)=0[/tex]
[tex]\implies (3u-4)(u+1)=0[/tex]
Apply the zero-product property:
[tex]\implies 3u-4=0 \implies u=\dfrac{4}{3}[/tex]
[tex]\implies u+1=0 \implies u=-1[/tex]
Substitute u=log(x) back in:
[tex]\implies \log x = -1[/tex]
[tex]\implies \log x= \dfrac{4}{3}[/tex]
On the left side of the original equation, we have log₃ (log x).
As logs of negative numbers cannot be taken, log(x)=-1 is not a valid solution since this would give log₃(-1) which is undefined.
Therefore, the only valid solution is:
[tex]\implies \log x= \dfrac{4}{3}[/tex]
[tex]\implies x=10^{\frac{4}{3}}[/tex]
[tex]\implies x=\left(10^4\right)^{\frac{1}{3}}[/tex]
[tex]\implies x=10000^{\frac{1}{3}}[/tex]
[tex]\implies x=\sqrt[3]{10000}[/tex]
Note: If a log has no base, assume that the base is 10.
Related Questions
from a group of 10 people, a committee of 5 is to be formed. the committee consists of a chair, a secretary and 3 regular members. (a) how many dierent committees are possible?
Answers
The number of ways of selecting a committees of 5 members from a group of 10 people will be 252
Combination is a method to calculate different ways to select items from a set / group regardless of the order.
ⁿCₓ = n! / (n-x)! x!
Total number of people in group is 10
Number of people require to form committee is 5
By using Combination ,
¹⁰C₅ = 10! / (10 - 5 )! 5!
= 10×9×8×7×6×5! / 5! × 5!
Cancelling 5! from numerator and denominator
= 10×9×8×7×6 / 5!
expanding the factorial
= 10×9×8×7×6 / 5×4×3×2×1
= 252
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A roll of ribbon was 8m long karmila cut 6 pieces pf ribbon each length 2/3 to tie some present then cut the remaining ribbon into some peices each of length 3/4
a) how many peices of ribbon each 3/4 m did karmila have
b) what was the length of ribbon left over
Answers
a) Karmila have 5 pieces of ribbon each 3/4 m long
b) the length of ribbon left over = 1 m
A roll of ribbon was 8m long.
Karmila cut 6 pieces of ribbon each length 2/3
Total length of 6 pieces of ribbon having length 2/3 would be,
6 * (2/3) = 4 m
This means, after cutting 6 pieces of ribbon, the length of the remaining ribbon = 4 m
Then we need to cut the remaining ribbon into some pieces each of length 3/4
Let t be the number of pieces.
So, we get an equation
(3/4) * t = 4
t = 4 * (4/3)
t = 16/3
t = 5 1/3
t ≈ 5
This means, Karmila have 5 pieces of ribbon each 3/4 m long and the length of ribbon left over = 1m
Therefore, a) Karmila have 5 pieces of ribbon each 3/4 m long
b) the length of ribbon left over = 1 m
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Can you help me because I don’t really understand what do you mean you know I just won the answer it or give me the answer
Answers
What is five and eight hundreds in standard form
Answers
Answer:5.08
Step-by-step explanation:
A polygon has 7 sides. The interior angle measures are 130°, 118°, 135°, 128°, 126°, 134°, and x°. Find the value of x.
Answers
Answer:
Step-by-step explanation:
A septagon (7 sided polygon) adds up to 900 degrees. When adding up the given interior angle measures, you get 771. Subtract 771 from 900 and you get the answer of 129.
Answer:
x= 129°
Step-by-step explanation:
Interior angles: 180(n-2) where n=sides
=180(7-2)
=180(5)
=900°
130°+118°+135°+128°+126°+134°+x=900
771+x=900
x=900-771
x= 129°
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Which ordered pair is a solution of the linear equation system
x + 4y = 18 and -3x + 2y = 16?
F. ( -2,-5)
G. (-2, 5)
H. (2, 5)
I. (2, -5)
Answers
Answer:
G is correct.
Step-by-step explanation:
[tex] - 2 + 4(5) = - 2 + 20 = 18[/tex]
[tex] - 3( - 2) + 2(5) = 6 + 10 = 16[/tex]
So G (-2, 5) is correct.
6. Choose the correct answer.
Which theorem or postulate could be used to prove the congruence of the pair of triangles?
Answers
Answer: SAS (Side angle side)
Does this figure have symmetry?
Answers
Answer:
Yes, because if you fold it in half, both halves match up.
Step-by-step explanation:
You can fold the figure in half along the x-axis to get a "D" shape
the picture is my question
Answers
Answer:
[tex] \rm \implies \frac{1}{3} (5x - 8) + \frac{2}{3} x \\ \\ \rm \implies \frac{5}{3} x - \frac{8}{3} + \frac{2}{3} x \\ \\ \rm \implies \frac{5x + 2x}{3} - \frac{8}{3} \\ \\ \rm \implies \frac{7x}{3} - \frac{8}{3} \\ \\ \rm \implies 2 \frac{1}{3} x - 2 \frac{2}{3} [/tex]
What is the height of a trapezoid with an area of 24 square inches if the two bases measure 4 inches and 6 inches?.
Answers
4.8 inches is the height of a trapezoid with an area of 24 square inches if the two bases measure 4 inches and 6 inches.
Using the formula for the area of a trapezoid, which is:
Area = (1/2)(b1 + b2)h
In this formula, b1 and b2 are the lengths of the two bases of the trapezoid, and h is the height of the trapezoid. In our problem, we know that the area is 24 square inches, and the lengths of the two bases are 4 inches and 6 inches. We want to solve for h, the height of the trapezoid.
Plugging in the known values, we get:
24 = (1/2)(4 + 6)h
24 = (1/2)(10)h
24 = 5h
4.8 = h
Therefore, the height of the trapezoid is 4.8 inches.
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marcus borrows $3,000 from his local credit union, to be repaid with 3.5% simple interest over 4 years. What are the principal, interest rate, and time in this situation?
Answers
Interest rate=3.5
Time=4 years
=42k
FLE.
= a
=t.
18. The table shows the amount of fabric
Camille used based on the number of
headbands that she made. Find the slope, and
explain what it means in the context of the
situation.
Headbands
Fabric (yd)
3
3
5
5
1
7775
5015
9
q
Answers
The slope of the line is 1/5 and it means the each headband uses 1/5 yards of fabrics
First choose two points
The first point = (3, 3/5)
The second point = (5, 1)
The slope of the line is the change in y coordinate with respect to the change in x coordinate of the line
The slope of the line = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
Where [tex](x_1,y_1)[/tex] is the coordinates of the first point
[tex](x_2,y_2)[/tex] is the coordinates of the second point
Substitute the values in the equation
The slope of the line = (1- 3/5) / (5 -3)
= (2/5) / 2
= 1/5 yards per headband
Hence, the slope of the line is 1/5 and it means the each headband uses 1/5 yards of fabrics
The complete question is
The table shows the amount of fabric Camille used based on the number of headbands that she made. Find the slope, and explain what it means in the context of the situation.
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An infant drinks about 26 ounces of milk per day. About how many quarts of milk does the baby drink in 4 weeks?
Answers
Answer:
Expert-Verified Answer
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Answer: 19.25 quarts of milk the baby drink in 4 weeks.
We know 1 week = 7 days
So, 4 weeks = 4*7 = 28 days.
In 1 day baby drinks = 22 ounce
In 28 days baby drinks = 28*22 = 616 ounce.
1 quarts = 32 ounces.
So, 616 ounce = 616/32 = 19.25 quarts.
Answer:22.75
Step-by-step explanation: 26 x 7 x 4 = 728 ounces in 4 weeks
theres 32 ounces in a quart
so 728 over 32 will equal 22.75
a particular fruit's weights are normally distributed, with a mean of 641 grams and a standard deviation of 23 grams. if you pick 25 fruits at random, then 15% of the time, their mean weight will be greater than how many grams? give your answer to the nearest gram.
Answers
If you pick 25 fruits at random, then 15% of the time, their mean weight will be greater than 665 grams.
What is mean?
In statistics, mean is the ratio of sum of all the observations and total number of observations in a data set.
How to calculate their mean weight will be greater than how many grams?
Given that from the question
Mean = 641 grams
standard deviation 24 grams
If you pick 25 fruits at random, then 15% of the time, their mean weight will be greater than how many grams?
The formula for the calculating the x value in the normal distribution is,
x = μ + zσ
The level of significance is = 15% = 0.15
Therefore,
z = (=normsinv (1 - α))
= (=normsinv (1-0.15))
= (=normsinv(0.85))
= 1.036
The critical value of z is 1.036
z = 1.036
(x - μ) / σ = 1.036
x = μ + 1.036σ
= 641 + 1.036*23
= 641 + 23.828
= 664.828
= 665grams (rounded to the nearest gram)
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An object is dropped from the top of a tower with a height of 1190 feet. Neglecting air resistance, the height of the object at time t seconds is given by the polynomial "-16t^2+1190." Find the height of the object at t=1 second
Answers
The height of the object at the time t = 1 second is given by 1174 feet.
Given the height of tower is 1190 feet.
The height of the object at time t seconds is given by,
h(t) = [tex]-16t^2[/tex] + 1190
So the height of the object after time t = 1 second is given by,
h(1) = [tex]-16\times 1^2+1190[/tex] = -16 + 1190 = 1174 feet
Hence the height of the object at the time t = 1 second is 1174 feet.
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You fill up your car with 15 gallons of gasoline. How many pounds does the gasoline weigh?
Hint: (15 gal) (x qt/gal) (y pt/ qt) (z lb/pt)
Answers
When You fill up your car with 15 gallons of gasoline. then it weighs 90 pounds of weight.
What is Weight?the weight of an object is the force acting on the object due to gravity.
Given
A car with 15 gallons of gasoline is filled.
We need to find the weight of the gasoline
We know that 1 gallons of gasoline is 6 pounds
So now for 15 gallons will be fifteen times of 6 pounds
15×6
90 pounds
90 pounds is the weight of the gasoline.
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Draw the given optimization problem and solve it. Find the volume of the largest right circular cylinder (in units3) that fits in a sphere of radius 4 units.
Answers
The volume of the largest right cylinder that fits in a sphere of radius 4 units is 143 m².
Let the diameter of the cylinder be “d" and the height of the cylinder be “h".
Given radius of the sphere = r = 4m
Diameter of the sphere = d = 2r = 2×4 = 08 m
As the cylinder is inscribed in the sphere
Therefore, the diameter of the cylinder be “d" is equal to the height of the cylinder be “h".
i.e, d = h
By Pythagoras Theorem, In triangle ABC
AB²+BC²=AC²
Or, d²+h² = (8)²
Or, h²+h² = 64
Or, 2h² = 64
Or, h² = 64/2
Or, h = √32
And d =√32
Or, r = d/2
= √32/2 =16/√32
Or, r = 16/√32
Volume of the cylinder = πr²h
=22/7 × (16/√32 )² × √32
= 22/ 7 × 8 ×√32
= 142.229
Rounding up the value 142.229 in 143 m³
Therefore, the largest volume of the cylinder = 143 m³
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A canoe rental service charges a $20 transportation fee and $30 dollars an hour to
rent a canoe. Write and graph an equation representing the cost, y, of renting a
canoe for × hours.
Answers
The equation representing the cost of the canoe rental company is given by y = 30x + 20 .
Let the total cost for renting a canoe be y.
Let the canoe is being rent for x hours.
now the cost for renting the canoe at 30 dollars per hour = 30x
Total cost of renting the canoe = 30x + 20
But the total cost is y.
Hence the equation to represent this situation is given by
y = 30x + 20 which is a straight line of the form y = mx +c, where the slope is 30 and the y-intercept is 20.
Now we will plot the graph for this equation.
At x = 0 , y = 20
At x = 1 , y = 50
At x= -3 , y = -70
Hence the graph of the line passes through these points on the cartesian plane.
The graph of the line is attached below.
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fill in the missing field to complete the program. determine the number of elements in the list that are divisible by 10. (hint: the number x is divisible by 10 if x % 10 is 0.) div ten
Answers
The number of elements in the list that are divisible by 10 is 10, 20, 30, 1000, 5000, 60000, etc.
Given:
the number of elements in the list that are divisible by 10. (hint: the number x is divisible by 10 if x % 10 is 0.) div ten
The number is divisible by 10 and the last digit is zero.
suppose take 10:
= 10/10 = 1
It is divisible.
= 10%10
The last digit or remainder is zero.
Therefore the number of elements in the list that are divisible by 10 is 10, 20, 30, 1000, 5000, 60000, etc.
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There's nothing here
Answers
Evaluate the expression if a=−12, b=34, and c=−23.
−||6c−16b||+1
Answers
The value of the expression − || 6c − 16b || + 1 if a = −12, b = 34, and c = −23 is - 681.
We that the modulus always gives a positive quantity.
For example: || -5 || = 5, etc.
We are given;
a = −12, b = 34, and c = −23.
We need to find − || 6c − 16b || + 1.
The solution of an algebraic expression is obtained by substituting the given values in the expression.
Substitute the given value in the above expression, we will get;
− || 6c − 16b || + 1
= − || 6 * -23 − 16 * 34 || + 1
= − || - 138 − 544 || + 1
= − || - 682 || + 1
= − 682 + 1
= - 681
So, the value of − || 6c − 16b || + 1 is - 681.
Thus, the value of the expression − || 6c − 16b || + 1 if a = −12, b = 34, and c = −23 is - 681.
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A painter can paint 175 square feet per hour. She charges $0.54 per square foot. Today she paints for 5.5 hours. How much will she charge?
Answers
The amount that the painter will charge is $519.75.
What is the cost?From the information illustrated, the painter can paint 175 square feet per hour and she charges $0.54 per square foot and then paints for 5.5 hours.
The amount that she will charge will be:
= Number of square foot × Amount charged × Number of hours
= 175 × $0.54 × 5.5
= $519.75
The amount is $519.75.
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A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of 1.15% per day. Write an exponential model representing the amount of Iodine-125 remaining in the tumor after t days. Then use the formula to find the amount of Iodine-125 that would remain in the tumor after 60 days. Enter the exact answer. Enclose arguments of functions in parentheses and include a multiplication sign between terms. For example, c* exp (h) or c * In (h). ab sin(a) 8 22 R ar A(t) == Round to the nearest tenth of a gram. There will be Number grams of Iodine-125 after 60 days.
Answers
Exponential model representing the amount of Iodine-125 remaining in the tumor after t days is f(t) = 0.5 *[tex]e^{-0.0115t}[/tex] and number grams of Iodine-125 after 60 days is 0.2507 grams
Given:
A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of 1.15% per day.
The formula for exponential decay is f(t)=a*e^kt where a is the original amount of the substance, k is the decay rate and t is the time elapsed. You are given the decay rate of 1.15% or -0.0115. For this problem t is left as is since we are creating a function of time.
f(t) = 0.5 *[tex]e^{-0.0115t}[/tex]
After 60 days
= 0.5 * [tex]e^{-0.0115*60}[/tex]
= 0.5*e^-0.69
= 0.5*0.50157
f(t) = 0.2507 grams
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The figure below is dilated by a factor of centered at the origin. Plot the resulting
image.
Click twice to plot a segment.
Click a segment to delete it.
Answers
The resulting dilation of the figure with the scale factor of 3/4 is given by the graph at the end of the answer.
Dilation on a image
Dilation in a image means multiply the coordinates of each vertex of the figure by the scale factors.
Given,
Here we have the triangle in the given figure.
Now, we have to find the dilation of the figure with the scale factor of 3/4.
Here from the original figure graph, the vertices are obtained as follows:
S(-8,-4), T(4, -4) and U(4,8)
Now we have to use the dilation scale factor of 3/4,
Which means that the coordinates are multiplied by 3/4,
Then the new coordinates of the resulting image are given as follows:
S(-8, -4) => ((-8 x 3/4), (-4,3/4)) => S'(-6, -3)
T(4, -4) => ((4 x 3/4), (-4,3/4)) => T'(3, -3)
U(8, 4) => ((8 x 3/4), (4,3/4)) => U'(6, 3)
Now, we have to plot these new coordinates on the graph, then we get,
the graph like the following.
Here the dilated figure is given by the graph at the end of the answer, having the same format as the original figure, just with smaller side lengths due to the dilation with a scale factor less than 1.
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Help really bad !!. I don’t understand for some reason
Answers
Answer:
x=-20
y=10
z=-19
Step-by-step explanation:
It is given that the shape is a rhombus, which means all four sides are equal. This means we can set up the equations:
5y-4=46 --> 5y=50 --> y=10
-2x+6=46 --> -2x=40 --> x=-20
-2z+8=46 --> -2z=38 --> z=-19
p: -3x + 8x - 5x = x
q: (3x)(5y) = 15xy
which of the following statements is false?
Answers
From the given statements it was found that statement p is false. So, -3x + 8x - 5x is equal to 0 not x.
How do equations work?A mathematical statement known as an equation is made up of two expressions joined together by the equal sign. An equation is, for instance, 3x - 5 = 16.In p statement,
-3x + 8x - 5x = x is given.
Let us check it.
-3x + 8x - 5x = 8x - 8x = 0
So, we can say statement p is false.
In q statement,
(3x)(5y) = 15xy is given.
Let us check it.
(3x)(5y) = 15xy
So, we can say statement is q is true.
Therefore, from the given statements it was found that statement p is false. So, -3x + 8x - 5x is equal to 0 not x.
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A ratio is a comparison of two quantities. For example, the ratio of cows to pigs on a farm could be 1 to 3. For every 1 cow, there are 3 pigs. The ratio of chocolate chips to raisins in a cookie could be 5 to 9. For every 5 chocolate chips in the cookie, there are 9 raisins. Do you notice a repeating phrase in these examples? “For every” is key ratio language. It tells you how much of one quantity exists in comparison to the other quantity. A ratio consists of just the numbers, not the units. If someone asked the ratio of chocolate chips to raisins in the cookie example, you’d say 5 to 9.
Which of these could you represent with a ratio?
Answers
Answer:vvvvvvvvvvvvvvvvvv55555
Step-by-step explanation:
if f(x) = -6x - 2. find f (-3)
Answers
Answer:
f(- 3) = 16
Step-by-step explanation:
substitute x = - 3 into f(x) , that is
f(- 3) = - 6(- 3) - 2 = 18 - 2 = 16
Olivia is gathering sugar for a cookie fundraiser for her soccer team. Olivia needs to gather at least 54 1/3 cups of sugar. She has already collected 15 2/3 cups of sugar. If she can collect 3 1/3 cups of sugar per day, how many more days will it take her to collect at least 54 1/3 cups of sugar?
Answers
The number of days that it take Olivia to collect at least 54 1/3 cups of sugar is 12 days.
What is an inequality?Inequalities are created through the connection of two expressions. It should be noted that the expressions in an inequality aren't always equal. They are denoted by the symbols ≥ < > ≤.
In this case, Olivia needs to gather at least 54 1/3 cups of sugar.
The expression will be illustrated thus:
15 2/3 + 3 1/3d ≥ 54 1/3
Collect like terms
3 1/3d ≥ 54 1/3 - 15 2/3
3 1/3d ≥ 38 2/3
Divide
d ≥ 38 2/3 ÷ 3 1/3
d ≥ (116 / 3 × 3/10)
d ≥ 11 3/5
She needs 12 days.
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A group of people walked 62 miles in 1 week and 58 miles in week 2. in week three they waled 14 miles less during week 2. how many miles did the group walk in 3 weeks
Answers
Answer:
Step-by-step explanation:
The of miles the group walked in 3 weeks is 44 miles
What is proportion?
Proportion can be defined as the mathematical comparison of two elements or numbers.
From the information given, we have that;
• in 1 week people walked 62 miles
• in the second week reduced to 58 miles
• in week 3 they walked 14 fewer miles during week 2
to determine the many miles did the group walk in 3 weeks, we can reduce
58 miles - 14 miles less during week 2
=44 miles.
so, number of miles the group walked in 3 weeks is 44 miles.