The first term of the geometric sequence is a(1) = 7.
The nth term of a geometric sequence is given by:
a(n) = a(1) . rⁿ⁻¹
Where:
a(1) = the first term
r = ratio
Parameter given in the problem:
a(5) = 112
Plug n = 5 into the formula
a(1) . r⁵⁻¹ = 112
a(1) . r⁴ = 112 ....... (1)
a(7) = 448
Plug n = 7 into the formula
a(7) = a(1) . r⁶
448 = a(1) . r⁶ ..... (2)
Divide (2) by (1)
448 : 112 = r⁶ : r⁴
4 = r²
Since r² = 4, then r⁴ = ( r² )² = 4² = 16.
Substitute r⁴ = 16 into equation (1):
a(1) . 16 = 112
a(1) = 112/16 = 7
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What are the solutions of the rational equation?
b. x/x+1 + 3 / x+4 = x+3 / x+4
The solution of the given rational equation [tex]\frac{x}{x+1}+\frac{3}{x+4}=\frac{x+3}{x+4}[/tex] is x = 0
In this question, we have been given a rational equation.
[tex]\frac{x}{x+1}+\frac{3}{x+4}=\frac{x+3}{x+4}[/tex]
We need to find the solution of the above rational equation.
[tex]\frac{x}{x+1}+\frac{3}{x+4}\\\\=\frac{x(x+4)+3(x+1)}{(x+1)(x+4)}\\\\ =\frac{x^{2}+4x+3x+3 }{(x+1)(x+4)} \\\\=\frac{x^{2} +7x+3}{(x+1)(x+4)}[/tex]
So given equation would be,
[tex]\frac{x^{2} +7x+3}{(x+1)(x+4)}=\frac{x+3}{x+4}\\\\x^{2} +7x+3=(x+1)(x+3)\\\\x^{2} +7x+3=x^{2} +x+3x+3\\\\x^{2} +7x+3=x^{2} +4x+3\\\\7x-4x=3-3\\\\3x=0\\\\x=0[/tex]
Therefore, the solution of the given rational equation [tex]\frac{x}{x+1}+\frac{3}{x+4}=\frac{x+3}{x+4}[/tex] is x = 0
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Dan spends 1/5 of his wages on rent and 2/3 on food. If he makes £495 per week, how much money does he have left?
Dan is left with £66 after the total spending.
What is referred as the total expenses?An expense is the amount of operations incurred by a business in order to generate revenue.
It is simply described as the amount of money required to obtain something. "It costs to make money," as the adage goes.Expenses can be classified in a number of ways. Fixed expenses, including such rent or mortgage, are those that don't change in response to changes in production. Expenses can also be described as variable expenses, which change in response to changes in production. Utilities as well as the cost of selling goods are examples of these.As for the given question;
The total earning per week is £495.
Spending on rent is 1/5.
1/5 of £495 = 495/5 = 99
Spending on rent is £99.
Now, spending on food is 2/3 of £495.
Food spending = (2/3)×495 = £330
Let the money left is 'x'.
Total expenses = Total income
food cost + rent cost + saving = 495
99 + 330 + x = 495
x = 495 - 429
x = 66
Thus, the savings by the Dan is £66.
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Find the volume of the solid generated by revolving the region bounded by the given curve and lines about the x-axis. y=5x, y=5, x=0
The volume of the solid generated by revolving the region bounded by the given curve and lines about the x-axis is V = 8π/3
What is the Volume of a solid?The volume of a solid, given by the function f(x), over an interval between a and b, is given by:
[tex]V = \pi \int\limits^a_b {f(x)^2} \, dx[/tex]
To find the area, integrate around that area. We're revolving around the x-axis so the area will be a circle;
V = ∫A(x)dx = ∫(πr²)dr
Then we subtract them from each others.
∫(πr₂² - πr₁²)dr
Now substitute the value and integrate;
∫(π(2)² - π(2x)²)dr
∫(4π - 4πx²)dr
4π∫(1 - x²)dr
Now integrate from 0 to 1 since that's where our boundary is.
V = 4π∫(1 - x²)dr = 8π/3
Therefore, The volume of the solid generated by revolving the region bounded by the given curve and lines about the x-axis is V = 8π/3
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What is the solution to the given system of equations?
7a-12b = 10
-2a-4b = 12
pls help me i don’t understand
Answer:
a=−2 and b=−2
Step-by-step explanation:
Let's solve your system by substitution.
7a−12b=10;−2a−4b=12
Step: Solve7a−12b=10for a:
7a−12b=10
7a−12b+12b=10+12b(Add 12b to both sides)
7a=12b+10
7a/7=12b+10/7
(Divide both sides by 7)
[tex] \sf \: a = \frac{12}{7} b + \frac{10}{7} [/tex]
Step: Substitute: 12/7b+10/7 for a in−2a−4b=12:
−2a−4b=12
−2(12/7b+10/7)−4b=12
−52/7b + −20/7=12(Simplify both sides of the equation)
−52/7b+−20/7+20/7=12+20/7
(Add 20/7 to both sides)
−52/7b=104/7
[tex] \sf \: \frac{ \frac{ - 52}{7}b }{ \frac{ - 52}{7} } = \frac{ \frac{104}{7} }{ \frac{ - 52}{7} } [/tex]
(Divide both sides by (-52)/7)
b=−2
Step: Substitute −2 for b in a =12/7b+10/7:
a=/12/7b+10/7
a=12/7(−2)+10/7
a=−2(Simplify both sides of the equation)
Help me someoneeeeeeeeeeeeeeeeeeeeeeeee
Answer:
Step-by-step explanation:
Identify the pattern and find the next three terms.
50,45,40,35, .......
Simplify each sum or difference. State any restrictions on the variables.
5x/x² - x-6 + 4/x² + 4x + 4
The simplification of the given each sum or difference of the expression [tex]\frac{5x}{x^{2} -x-6} + \frac{4}{x^{2} +4x+4}[/tex] is equal to [tex]\frac{5x^{2} +14x-12}{(x-3)(x+2)^{2}}[/tex] , restriction on the variables is given by x ≠ 3 , x≠ -2.
As given in the question,
Given expression : [tex]\frac{5x}{x^{2} -x-6} + \frac{4}{x^{2} +4x+4}[/tex]
Simplify the given expression by taking least common multiple,
[tex]\frac{5x}{x^{2} -x-6} + \frac{4}{x^{2} +4x+4}\\\\\\= \frac{5x}{x^{2} -3x+2x-6} + \frac{4}{(x+2)^{2}} \\\\\\= \frac{5x(x+2)+4(x-3)}{(x-3)(x+2)^{2}}\\ \\=\frac{5x^{2} + 14x-12}{(x-3))x+2)^{2}}[/tex]
Restriction on the variables is given by x ≠ 3 , x≠ -2.
Therefore, simplification of the given each sum or difference of the expression [tex]\frac{5x}{x^{2} -x-6} + \frac{4}{x^{2} +4x+4}[/tex] is equal to [tex]\frac{5x^{2} +14x-12}{(x-3)(x+2)^{2}}[/tex] , restriction on the variables is given by x ≠ 3 , x≠ -2.
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A bin of 50 manufactured parts contains 3 defective parts and 47 nondefective parts. two parts are selected randomly without replacement. let a denote the event that the first part is defective. let b be the event that the second part. is defective. find the p(b)
The probability that the event that the second part is defective will be 0.0564.
How to calculate the probability?From the information, the bin of 50 manufactured parts contains 3 defective parts and 47 nondefective
Therefore, the probability of having a defective and non defective parts will be:
= 47/50 × 3/50
= 0.94 × 0.06
= 0.0564
Therefore, the probability that the event that the second part is defective will be 0.0564.
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Write an equation in slope-intercept form for each line described.
passes through (-2,2) , perpendicular to y=-5 x-8 .
The equation in slope-intercept form is x – 5y = -12
Given,
y = -5x – 8
This is in slope-intercept form, y = mx + b
Where, m is slope and b is y-intercept.
Here, m = -5
Perpendicular lines have slopes that are negative reciprocals, [tex]m_{p}[/tex] = -1/m
Here we need equation of line through (-2, 2) with slope -1/-5 = 1/5
y = (1/5)x + b
where (-2, 2) is on the line.
2 = 1/5(-2) + b
2 = -2/5 + b
b = 2 + 2/5 = 12/5
therefore, equation we need is:
y = 1/5x + 12/5
that is,
1/5x – y = -12/5
Multiplying 5 in both sides, we get
5(1/5x) – 5y = (-12/5)5
That is, the equation we need is:
x – 5y = 12
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make an outcome tree to show all possible combinations. you have three pairs of shorts four shirts, flip flops, and sneakers
The required number of combinations is 24.
Given that,
To make an outcome tree to show all possible combinations. you have three pairs of shorts four shirts, flip-flops, and sneakers.
In arithmetic, combination and permutation are two different ways of grouping elements of a set into subsets. In a combination, the components of the subset can be recorded in any order. In a permutation, the components of the subset are listed in a distinctive order.
Here,
flip flops
short 1 short 2 short 3
Shirt 1 shirt 2 shirts 3 shirts 4
Number of combinations = 12
sneakers
short 1 short 2 short 3
Shirt 1 shirt 2 shirts 3 shirts 4
Number of combinations = 12
Total number of combination = 12 + 12 = 24
Thus, the required number of combinations is 24.
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Alex earns a weekly base salary of $650 as a car salesman. He also earns a 9% commission on his monthly sales after the first $10,000. What was his total salary this month if he sold 7 cars whose average price was $4,500 each?
The total salary for the month will be $4535
How to calculate the value?From the information, Alex earns a weekly base salary of $650 as a car salesman. He also earns a 9% commission on his monthly sales after the first $10,000.
If he sold 7 cars whose average price was $4,500 each. The total amount will be:
= $4500 × 7.
= $31500
The commission will be
= 9% × ($31500 × $10000)
= 9% × $21500
= $1935
The total salary for the month will be:
= ($650 × 4) + $1935
= $4535
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The formula a=m-a represents the actual cost, a, of an item with original price m after a coupon for in dollars off is applied. Solve the formula for the amount of the coupon.
The formula a=m-a represents the actual cost, a, of an item with original price loss m after a coupon for in dollars off is applied.
n = m - a
In light of this:
The formula a = m - n's expression
where;
a = the item's real cost
m = initial cost
n = the monetary value of the coupon
The goal is to figure out how much the coupon is worth.
To calculate the quantity of the coupon, we must subtract n from the above statement, where n is the number of coupons in dollars.
As a result, if a = m - n
a + n Equals m
n = m - a
Profit and loss are expressions used to determine whether or not a sale is economic. we use these expressions frequently. However, the difference between the two is appertained to as profit, If the selling price is advanced than the cost price. However, the difference is appertained to as the loss, If the selling price is lower than the cost price. The cost price of a product is the price at which it's bought.
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The addition expression below represents which of the following multiplications? 4+4+4
Answer:
4 × 3
Step-by-step explanation:
4 is being added 3 times.
find the value of - |3.4|
Answer:
-3.4
Step-by-step explanation:
"| |" are absolute value which means |-4| is actually 4. However, if there is a negative outside of the lines, than the answer is negative.
Carl writes 14/3 x 7/3 to find the quotient of 4 2/3 divided by 2 1/3. What is his mistake?
The quotient in this Carl multiplies the both value instead of dividing so that is the mistake that made by Carl.
The quotient of 4 divided by 2 1/3 and also explain the mistake that done by Carl
The value of 4 1/3 is 14/3
And the value of 2 1/3 is 7/3
So then we divide 4 2/3 by 2 1/3 is,
[tex]\frac{14 / 3}{7 / 3}=\frac{14}{3} \frac{3}{7}[/tex]
=2
Carl's mistake is thinking that his quotient (2) is equal to the value of the terms involved in it. In fact, 2 is just a multiplier for 7/3 and doesn't have any specific value associated with it.
Hence, this is the quotient in this Carl multiplies the both value instead of dividing so that is the mistake that made by Carl.
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Use the properties of logarithms to evaluate each expression. 3log₉ 3- 1/4 log₉ 81
The evaluation of each expression is 3/2 , 1/2.
What is logarithms:
Logarithms are the other way of writing the exponents. A logarithm of a number with a base is equal to another number. A logarithm is just the opposite function of exponentiation. For example, if 102 = 100 then log10 100 = 2.
Hence, we can conclude that,
Log b x = n or bn = x
Where b is the base of the logarithmic function.
This can be read as “Logarithm of x to the base b is equal to n”.
In this article, we are going to learn the definition of logarithms, two types of logarithms such as common logarithm and natural logarithm, and different properties of logarithms with many solved examples.
What is properties of logarithms:
Properties of logarithms functions are used to solve logarithm problems. We have learned many properties in basic math's such as commutative, associative and distributive, which are applicable for algebra. In the case of logarithmic functions, there are basically five properties.
This are as follows:
The product property
The quotient property
Power rule
Change of base rule
Reciprocal rule
Now,
3log₉ 3 = [(9)^x]^3 = 3
[(3^2)^x]^3 = 3
x = 3/2
Second part,
1/4 log₉ 81
= 1/2
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7) Krystal wants to make a 54% sugar
solution. She has already poured 2 qt. of a
62% sugar solution into a beaker. How
many qt. of a 38% sugar solution must she
add to this to create the desired mixture?
we know she poured already 2 qts of a 62% sugar solution, hmmm how much will that be in quarts? well, just (62/100) * 2 = 1.24 quarts.
we also know she wants to mixed say "x" quarts of a 38% sugar solution, how many quarts will that be? well, (38/100) * x = 0.38x.
let's make a quick table from that as well as including the desired mixture.
[tex]\begin{array}{lcccl} &\stackrel{solution}{quarts}&\stackrel{\textit{\% of }}{amount}&\stackrel{\textit{qts of }}{amount}\\ \cline{2-4}&\\ \textit{62\% solution}&2&0.62&1.24\\ \textit{38\% solution}&x&0.38&0.38x\\ \cline{2-4}&\\ desired~mixture&y&0.54&0.54y \end{array}\qquad \begin{cases} ~\hfill 2+x&=y\\\\ 1.24 + 0.38x &= 0.54y \end{cases} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{\textit{substituting on the 2nd equation}}{1.24 + 0.38x ~~ = ~~ 0.54(\underset{y}{2 + x})}\implies 1.24+0.38x=1.08+0.54x \\\\\\ 0.16+0.38x=0.54x\implies 0.16=0.16x\implies \cfrac{0.16}{0.16}=x\implies \stackrel{qts}{1}=x[/tex]
Plss help me A scale drawing of a lake has a scale of 2 cm to
100 m. If the actual width of the lake is 1,500 m, what is
the width of the lake on the scale drawing?
The width of the lake on the scale drawing is 30 cm.
Based on the given conditions, formulate 1500*2/100
Cross out the common factor: 15*2 = 30
Get the result as 30 cm
An object is enlarged in a scale drawing. An expansion increases or decreases the size of an object by multiplying each of its lengths by a scale factor. Typically, a ratio is used to describe the scale of a drawing. a representation of a real thing with correct sizes that have been scaled back or blown up slightly (called the scale). The scale is represented by the length in the drawing, a colon (":"), and then the actual length that corresponds to it.
Hence, the width of the lake on the scale drawing is 30 cm.
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points!!!!!!!!!!!!!!!!
Answer:
5 points
Step-by-step explanation:
because yes
Draw a tessellation using the following shape(s).
trapezoid and a parallelogram
Tessellation in trapezoid is possible only if it is rotated
see the fig, attached.
But tessellation in parallelogram is possible both rotated and non rotated
see the fig, attached.
What is tessellation?A tessellation or tiling is the seamless application of one or more geometric shapes, known as tiles, to a surface, frequently a plane. Tessellation in mathematics can be expanded to include other geometries and higher dimensions.
A tiling constructed of materials like cemented ceramic hexagons or squares is referred to as a real-world tessellation. Such tilings may be ornamental designs or serve practical purposes such providing long-lasting and water-resistant pavement, floor, or wall coverings.
Tessellations were historically employed in Islamic art, such as the Moroccan architecture and the ornamental geometric tiling of the Alhambra palace. They were also used in Ancient Rome.
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I need this answered please
Answer: 23 + 2c
Step-by-step explanation:
To write an expression, we will look at the wording of the phrase given;
The sum of ➜ addition
twice Chau's height ➜ 2c
The sum of 23 and twice Chau's height.
23 + 2c
What is 3/4 divided by 5 as a reciprocal
Answer:
I think its 3 20.
Step-by-step explanation:
Use the given information to write an equation of a circle.
center at (-1,-5) , radius 2
Using the given information, the equation of the circle with radius 2 and center at (-1 , -5) is (x + 1)^2 + (y + 5)^2 = 4.
The general form of the equation of circle is given by (x - h)^2 + (y - k)^2 = r^2, where (h , k) is the location of the center and r is the radius of the circle.
Given the radius and center of the circle, substitute these values to the general form of the equation of the circle.
(x - h)^2 + (y - k)^2 = r^2
where (h , k) = (-1 , -5)
r = 2
(x - -1)^2 + (y - -5)^2 = 2^2
(x + 1)^2 + (y + 5)^2 = 4
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A scale drawing of a school bus has a scale of ; inch to 6 feet. If the length of the school bus is 4, inches on the scale drawing, what'is the actual length of the bus?
A scale drawing of a school bus has a scale of ; inch to 6 feet. If the length of the school bus is 4, inches on the scale drawing, what'is the actual length of the bus?
Express the quantity as a positive or negative number the record low temperature for a date is 21.7 degrees below zero
The temperature will be represented by the negative number -21.7 degrees.
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that the quantity is a positive or negative number the record low temperature for a date is 21.7 degrees below zero.
T = 0 - 21.7
T = -21.7 degrees
Therefore, the temperature will be represented by the negative number -21.7 degrees.
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What number equal to 48
Answer:
The factor pairs of the number 48 are: 1 x 48, 2 x 24, 3 x 16, 4 x 12, and 6 x 8. When you multiply each of these factor pairs together, you get 48.
Step-by-step explanation:
1 x 48 = 48
2 x 24 = 48
3 x 16 = 48
4 x 12 = 48
6 x 8 = 48
HOPE IT'S HELP:)
Determine whether each sequence is geometric. If so, find the common ratio. 7,0.7,0.07,0.007, . . . . .
The given sequence is geometric, with common ratio = 0.1.
What is a geometric progression?A geometric progression, sometimes referred to as a geometric sequence in mathematics, is a series of non-zero numbers where each term following the first is obtained by multiplying the preceding one by a constant, non-zero value known as the common ratio.
Given: 7, 0.7, 0.07, 0.007,...
Here, 0.7/7 = 0.07/0.7 = 0.007/0.07 = 0.1
That is, the given sequence follows the only necessary condition required for a sequence to be geometric.
Hence, The given sequence is geometric, with common ratio = 0.1.
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someone please help!
The model shows Earth and the Sun.
(picture below)
Why do different parts of Earth receive different amounts of sunlight during the day?
Find the number of possible outcomes for following situation.
Three number cubes are rolled simultaneously.
Experiment: Rolling three dice simultaneously
No of possible outcomes: 6*6*6 = 216
The Sample space is shown below:
(1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5) (1,1,6)
(1,2,1) (1,2,2) (1,2,3) (1,2,4) (1,2,5) (1,2,6)
(1,3,1) (1,3,2) (1,3,3) (1,3,4) (1,3,5) (1,3,6)
(1,4,1) (1,4,2) (1,4,3) (1,4,4) (1,4,5) (1,4,6)
(1,5,1) (1,5,2) (1,5,3) (1,5,4) (1,5,5) (1,5,6)
(1,6,1) (1,6,2) (1,6,3) (1,6,4) (1,6,5) (1,6,6)
(2,1,1) (2,1,2) (2,1,3) (2,1,4) (2,1,5) (2,1,6)
(2,2,1) (2,2,2) (2,2,3) (2,2,4) (2,2,5) (2,2,6)
(2,3,1) (2,3,2) (2,3,3) (2,3,4) (2,3,5) (2,3,6)
(2,4,1) (2,4,2) (2,4,3) (2,4,4) (2,4,5) (2,4,6)
(2,5,1) (2,5,2) (2,5,3) (2,5,4) (2,5,5) (2,5,6)
(2,6,1) (2,6,2) (2,6,3) (2,6,4) (2,6,5) (2,6,6)
(3,1,1) (3,1,2) (3,1,3) (3,1,4) (3,1,5) (3,1,6)
(3,2,1) (3,2,2) (3,2,3) (3,2,4) (3,2,5) (3,2,6)
(3,3,1) (3,3,2) (3,3,3) (3,3,4) (3,3,5) (3,3,6)
(3,4,1) (3,4,2) (3,4,3) (3,4,4) (3,4,5) (3,4,6)
(3,5,1) (3,5,2) (3,5,3) (3,5,4) (3,5,5) (3,5,6)
(3,6,1) (3,6,2) (3,6,3) (3,6,4) (3,6,5) (3,6,6)
(4,1,1) (4,1,2) (4,1,3) (4,1,4) (4,1,5) (4,1,6)
(4,2,1) (4,2,2) (4,2,3) (4,2,4) (4,2,5) (4,2,6)
(4,3,1) (4,3,2) (4,3,3) (4,3,4) (4,3,5) (4,3,6)
(4,4,1) (4,4,2) (4,4,3) (4,4,4) (4,4,5) (4,4,6)
(4,5,1) (4,5,2) (4,5,3) (4,5,4) (4,5,5) (4,5,6)
(4,6,1) (4,6,2) (4,6,3) (4,6,4) (4,6,5) (4,6,6)
(5,1,1) (5,1,2) (5,1,3) (5,1,4) (5,1,5) (5,1,6)
(5,2,1) (5,2,2) (5,2,3) (5,2,4) (5,2,5) (5,2,6)
(5,3,1) (5,3,2) (5,3,3) (5,3,4) (5,3,5) (5,3,6)
(5,4,1) (5,4,2) (5,4,3) (5,4,4) (5,4,5) (5,4,6)
(5,5,1) (5,5,2) (5,5,3) (5,5,4) (5,5,5) (5,5,6)
(5,6,1) (5,6,2) (5,6,3) (5,6,4) (5,6,5) (5,6,6)
(6,1,1) (6,1,2) (6,1,3) (6,1,4) (6,1,5) (6,1,6)
(6,2,1) (6,2,2) (6,2,3) (6,2,4) (6,2,5) (6,2,6)
(6,3,1) (6,3,2) (6,3,3) (6,3,4) (6,3,5) (6,3,6)
(6,4,1) (6,4,2) (6,4,3) (6,4,4) (6,4,5) (6,4,6)
(6,5,1) (6,5,2) (6,5,3) (6,5,4) (6,5,5) (6,5,6)
(6,6,1) (6,6,2) (6,6,3) (6,6,4) (6,6,5) (6,6,6)
Favorable outcome (three similar numbers, i.e. getting an identical number in all the three dice) : 6
Probability = No of favorable outcomes/Total no of outcomes = 6/216 = 0.0278
The perimeter of a rectangle is 30cm, it’s width is 6cm. Find its area
Step-by-step explanation:
the area of a rectangle is
length × width.
so, in our case
30 × 6 = 180 cm²
be careful to use the right unit for the result. an area is measured in square units based on the units of the given lengths. hence cm² as the given lengths are in cm.
