Equation [1] : a(n) = 3 + (n - 1)5 represents the step where he made the mistake.
We have,
A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.
We have a student who used the explicit formula a[n] = 5+3(n-1) for the sequence 3,8,13,18,23,...to find the 12th term.
The given sequence is -
3 , 8 , 13 , 18 , 23, ..
Here -
d = 8 - 3 = 13 - 8 = 5
a = 3
a(n) = a + (n - 1)d {for arithmetic sequence}
a(n) = 3 + (n - 1)5 ...Eq[1]
a(n) = 3 + 5n - 5
a(n) = 5n - 2
Therefore, Equation [1] : a(n) = 3 + (n - 1)5 represents the step where he made the mistake.
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complete question:
A student uses the explicit formula an= 5+3(n-1) for the sequence 3,8,13,18,23,...to find the 12th term. Explain the error the student made.
Cody works at a vehicle rental company. The vehicles available to rent are listed in the table.
What is the ratio of seats to engine cylinders in the truck?
The Ratio of seats to engine cylinders in the Ford F-150 truck that Cody wants to rent is approximately 0.625.
The ratio of seats to engine cylinders in the Ford F-150 truck that Cody wants to rent can be calculated by dividing the number of seats by the number of engine cylinders. In this case, the Ford F-150 has a seating capacity of 5 passengers and a V8 engine configuration.
To calculate the ratio, we divide the number of seats (5) by the number of engine cylinders (8):
Ratio of seats to engine cylinders = Number of seats / Number of engine cylinders
Ratio of seats to engine cylinders = 5 / 8
Simplifying the ratio, we get:
Ratio of seats to engine cylinders = 0.625
Therefore, the ratio of seats to engine cylinders in the Ford F-150 truck that Cody wants to rent is approximately 0.625.
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Note the full question may be :
Cody works at a vehicle rental company. The vehicles available to rent are listed in the table below:
Vehicle Type | Seats | Engine Cylinders
----------------------------------------------
Car | 5 | 4
SUV | 7 | 6
Truck | 3 | 8
What is the ratio of seats to engine cylinders in the truck?
A. 1:3
B. 3:8
C. 8:3
D. 3:1
Choose the correct option that represents the ratio of seats to engine cylinders in the truck.
The cost of 4 pairs of shorts and 3 sweatshirts is $89. The cost of 9 pairs of shorts and 11 sweatshirts is $247
Enter the cost of one pair of shorts and one sweatshirt
Find the y-intercept of the line 12x - 18y = -24.
Answer:
Therefore, the y-intercept of the line is (0, 4/3).
Step-by-step explanation:
To find the y-intercept of the line 12x - 18y = -24, we need to set x = 0 and solve for y.
12(0) - 18y = -24
Simplifying, we get:
-18y = -24
Dividing both sides by -18:
y = 4/3
Therefore, the y-intercept of the line is (0, 4/3).
LIGHT The length of the shadow S given by a tower
that is 100 meters high is S = 100/tan θ, where θ
is the angle of inclination of the sun. If the
angle of inclination is 45°, find the length of the shadow.
a. 162 m
b. 62 m
c. 100 m
d. 84 m
Answer:
c. 100 m
Step-by-step explanation:
You want the length of the shadow of a 100 m tower if the angle of inclination of the sun is 45°.
Isosceles right triangleThe tower and its shadow are modeled as legs of a right triangle. When the acute angle in such a triangle is 45°, the triangle is isosceles and the legs are the same length.
The shadow is the same length as the height of the tower: 100 m.
__
Additional comment
You can check this by using the formula:
S = 100/tan(45°) = 100/1 = 100
<95141404393>
estimate the original volume of the pyramid in fig 15.28 given that it's frustum has a 4 m by 4 m squared top which is 12 m vertically above the square base which is 20 m by 20 m (assume that the original problem was raised to a point . Neglect the volume of the entrance of the right of the photograph)
Answer:
Step-by-step explanation:
To estimate the original volume of the pyramid, we can use the formula for the volume of a frustum of a pyramid:
V = (1/3)h(a^2 + ab + b^2)
Where V is the volume, h is the height of the frustum, a is the side length of the top square, and b is the side length of the bottom square.
In this case, the top square has a side length of 4 m, and the bottom square has a side length of 20 m. The height of the frustum is 12 m.
Plugging these values into the formula, we get:
V = (1/3) * 12 * (4^2 + 4*20 + 20^2)
Simplifying the equation:
V = (1/3) * 12 * (16 + 80 + 400)
V = (1/3) * 12 * 496
V = 12 * 496/3
V = 1984
Therefore, the estimated original volume of the pyramid is 1984 cubic meters.
Find the values of c guaranteed by the Mean Value Theorem (MVT) for f(x)=……..
Using the mean value theorem in f(x) = 1/2x² + 7, c = 2√3
What is the mean value theorem?The mean value theorem states that let f be continuous over the closed interval [a,b] and differentiable over the open interval (a,b). Then, there exists at least one point c ∈(a,b) such that f'(c) = [f(b) - f(a)]/(b - a)
To find c using the mean value theoren for f(x) = 1/2x² + 7 over the interval [0, 6], we proceed as follows.
Uisng the mean value theorem, we know that
f'(c) = [f(b) - f(a)]/(b - a)
⇒ f(c) = 1/(b - a)∫ₐᵇ[f(x)
Now over the interval [a, b] = [0,6]
f(c) = 1/(6 - 0)∫₀⁶[f(x)
Now, since f(x) = 1/2x² + 7
Substituting this into the equation, we have that
f(c) = 1/(6 - 0)∫₀⁶[f(x)
f(c) = 1/(6 - 0)∫₀⁶(1/2x² + 7)
f(c) = 1/6∫₀⁶(1/2x² + ∫₀⁶7)
f(c) = 1/6[1/2x³/3 + 7x]₀⁶
f(c) = 1/6[x³/6 + 7x]₀⁶
f(c) = 1/6[6³/6 + 7(6)] - [0³/6 + 7(0)]
f(c) = 1/6[6²+ 42] - [0 + 7(0)]
f(c) = 1/6([36 + 42] - [0 + 0])
f(c) = 1/6(78 - 0)
f(c) = 1/6(78)
f(c) = 13
Now, since f(x) = 1/2x² + 7
f(c) = 1/2c² + 7
1/2c² + 7 = 13
1/2c² = 13 - 7
1/2c² = 6
c² = 2 × 6
c² = 12
c = √12
c = 2√3
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Find m/ABC.
A. 81°
B. 86°
C. 47°
D. 49.5°
Α.
99°
B
D
94%
C
m/ABC=86° (because of the cyclic quadrilaterals). Hence, Option (B) is correct.
A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a common circle.
The opposite angles of a cyclic quadrilateral have a total of 180°.
m/ABC + m/ADC = 180°
m/ABC + 94° = 180°
m/ABC = 180° - 94°
m/ABC = 86°
Similarly, one can find m/DCB
m/DAB + m/DCB = 180°
99° + m/DCB = 180°
m/DCB = 180° - 99°
m/DCB = 81°
Hence, m/ABC = 86°
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What is the volume of this figure?
Enter your answer in the box.
ft³
Three-dimensional figure formed by placing two rectangular prisms together to form an L shape where the bottom of the L is on the left and the top of the L extends to the right. The wider part of the L has a width of 3 feet. The other part of the L has a width of 2 feet. The total length of the L is 7 feet, and the length of the narrower part of the L is 6 feet. The height connecting the L shaped bases is 5 feet.
The volume of the rectangular prism in this problem is given as follows:
99 ft³.
How to obtain the volume of a rectangular prism?The volume of a rectangular prism, with dimensions defined as length, width and height, is given by the multiplication of these three defined dimensions, according to the equation presented as follows:
Volume = length x width x height.
The figure in this problem can be divided into two prisms, with dimensions given as follows:
3 ft, 5 ft and 7 - 6 = 1 ft.6 ft, 2 ft and 7 ft.Hence the volume of the prism is given as follows:
V = 3 x 5 x 1 + 6 x 2 x 7
V = 99 ft³.
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2/5 of Jim's savings is equal to 4/9 of Max's savings. Jim has $30 more than Max. 5 How much money does Max have?
As Jim has $30 more than Max, as per this, Max has $270.
Let's assume Max's savings as 'x' dollars.
We have:
2/5 of Jim's savings is equal to 4/9 of Max's savings.
(2/5) * Jim's savings = (4/9) * Max's savings
We also know that Jim has $30 more than Max.
Jim's savings = Max's savings + $30
So,
(2/5) * Jim's savings = (4/9) * Max's savings
(2/5) * (Max's savings + $30) = (4/9) * Max's savings
To solve for Max's savings, we can solve this equation:
(2/5) * (x + $30) = (4/9) * x
So, for x:
(2/5) * (x + $30) = (4/9) * x
Multiplying both sides by 45 (the least common denominator of 5 and 9) to eliminate the fractions:
18(x + $30) = 20x
18x + 540 = 20x
540 = 20x - 18x
540 = 2x
x = 270
Therefore, Max has $270.
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Evaluate the following statement.
( - 179 – 78 ) + 41 - ( - 165 )
The answer is:
( - 179 – 78 ) + 41 - ( - 165 )
=-51
Answer:
(-257) +41+165
-257+206
-51
A young artist went to an art shop to buy a canvas. There were many sizes from which to choose. Some were standard sizes and some were custom-made. She wanted her next piece to incorporate the Golden Ratio, so she decided to buy the canvas whose dimensions most closely matched the Golden Rectangle. Here are the sizes that were available:
24" x 36"
24" x 40"
10" x 16"
26" x 16"
20" x 12"
Which canvas should the artist buy? Show your work.
The artist should buy the canvas with dimensions 26" x 16" as it most closely matches the Golden Rectangle.
To determine which canvas size most closely matches the Golden Rectangle, we need to calculate the aspect ratios of the available options and compare them to the Golden Ratio.
The Golden Ratio is approximately 1.618. A rectangle is considered to be a Golden Rectangle if its aspect ratio (width divided by height) is equal to the Golden Ratio.
Let's calculate the aspect ratios for each canvas size:
Canvas 1: 24" x 36"
Aspect ratio = 24 / 36 = 0.67
Canvas 2: 24" x 40"
Aspect ratio = 24 / 40 = 0.6
Canvas 3: 10" x 16"
Aspect ratio = 10 / 16 = 0.625
Canvas 4: 26" x 16"
Aspect ratio = 26 / 16 = 1.625
Canvas 5: 20" x 12"
Aspect ratio = 20 / 12 = 1.667
Now, let's compare the aspect ratios of the available options with the Golden Ratio (1.618) to see which one is closest:
Canvas 1: |0.67 - 1.618| = 0.948
Canvas 2: |0.6 - 1.618| = 1.018
Canvas 3: |0.625 - 1.618| = 0.993
Canvas 4: |1.625 - 1.618| = 0.007
Canvas 5: |1.667 - 1.618| = 0.049
The canvas with the aspect ratio closest to the Golden Ratio is Canvas 4: 26" x 16". Its aspect ratio of 1.625 is only 0.007 away from the Golden Ratio.
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9*
Giovanni needs to hire a handyman to paint his house. Charles will charge him
$115 up front and $30 per hour. Peter will charge $40 per hour with an $80 flat fee
for paint. After how many hours will the two men charge the same amount to
complete the job?
The two men will charge the same amount to complete the job after 3.5 hours.
How to determine After how many hours will the two men charge the same amount to complete the job?Let's start with Charles' fee, which is a combination of an upfront fee and an hourly fee. We can represent this as:
C(x) = 115 + 30x
where x is the number of hours Charles works.
For Peter, his fee is a combination of an upfront fee for paint and an hourly fee. We can represent this as:
P(x) = 80 + 40x
where x is the number of hours Peter works.
To find out when the two men will charge the same amount to complete the job, we need to set the two equations equal to each other and solve for x:
115 + 30x = 80 + 40x
Subtracting 30x from both sides: 115 = 80 + 10x
Subtracting 80 from both sides: 35 = 10x
Dividing both sides by 10:
x = 3.5
So, the two men will charge the same amount to complete the job after 3.5 hours.
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Which expression is equivalent to 8.8x10^9/2.2x10^-3?
A 4 x 10^12
B 4 x 10^6
C 4 x 10^-3
D 4 x 10^-6
7. Given right triangle ABC below, determine sin(A).
The value of Sin A is 5/13.
Option A is the correct answer.
We have,
Sin A = Perpendicular / Hypotenuse
Sin A = BC / AB
And,
BC = 5
AB = 13
Substituting.
Sin A = 5/13
Thus,
The value of Sin A is 5/13.
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Students were asked their favorite ice cream flavor. The results showed that 196 students selected vanilla as their favorite ice cream flavor. This represents 49% of the total number of students surveyed. What was the total number of students surveyed?
I need the answer and explanation!!!
Answer:
The total number of students surveyed is 400.
Step-by-step explanation:
We want to know that 49% of what number is 196
Let x = the total number of students
.49x = 196 (49% as a decimal is .49) Divide both sides by .49
x = 400
50 Points! Multiple choice geometry question. Photo attached. Thank you!
Answer:
The correct answer:
(A) WZ/WX = XW/YW
Let's say you (a 16-year old) open a savings account with an interest rate of 6% per year and you
aren't adding any additional funds in the future. If you make $80,000 within the year you turn 60, what
is the total amount in your account at 60 years old?
The total amount in the account when turning 60 years is A = $ 11,13,706.08
Given data ,
A savings account with an interest rate of 6% per year and you aren't adding any additional funds in the future
Now , you make $80,000 within the year you turn 60
So , the number of years = 60 - 16 = 44 years
And , from the compound interest , we get
A = P ( 1 + r/n )ⁿᵇ
On simplifying , we get
Where A is the final amount, P is the initial amount (which is 0 in this case), r is the annual interest rate (6% or 0.06), n is the number of times the interest is compounded per year (let's assume it is compounded monthly, so n=12), t is the time in years (44 years from age 16 to age 60).
A = 80,000 ( 1 + 0.06/12 )¹²ˣ⁴⁴
A = $ 11,13,706.08
Hence , the amount in account is A = $ 11,13,706.08
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2/3x - 1/5x = x - 1
solve pls
Plssssss helllllp meeee 50.point and will mark Brainlyiest Solve: 4!
A.16
B.6
C.24
D.4
Answer:
C
Step-by-step explanation:
the expression ! ( factorial ) has the following meaning
n! = n(n - 1)(n - 2) ... × 3 × 2 × 1
Then
4! = 4 × 3 × 2 × 1 = 24
Urgent!!!
cos4α + 2 Sin² 2α
Answer:
Step-by-step explanation:
cos4α + 2 Sin² 2α is a trigonometric expression that involves the cosine and sine functions of the angle α. This expression can be simplified using trigonometric identities to get:
cos4α + 2 Sin² 2α = (cos² 2α)² + 2 (Sin 2α)²
This simplification involves using the double angle identity for sine and the identity for the square of cosine.
Brainliest Pls
Answer:1
Step-by-step explanation:
cos(4α) + 2sin²(2α)
First, let's expand cos(4α)
cos(4α) = cos²(2α) - sin²(2α)
Now, let's substitute this expression back into the original equation:
cos(4α) + 2sin²(2α) = (cos²(2α) - sin²(2α)) + 2sin²(2α)
= cos²(2α) + sin²(2α)=1 by
cos²(α) + sin²(α)=1
pls help it is due today
The property that should be used to simplify the expression 1/t⁻³ is [tex]\frac{1}{a^-m}[/tex] = [tex]a^m[/tex] (where a ≠ 0). Option A
How to identify the property to simplify the expression?In the given scenario or expression, this property implies that 1/t⁻³ = t³. The negative exponent in the denominator becomes a positive exponent when moved to the numerator.
Here is the step-by-step solution
1/t⁻³ = 1/(t⁻³)
1/(t⁻³) = (t⁻³)⁻¹
(t⁻³)⁻¹ = t⁽⁻³⁾ˣ⁽⁻¹⁾
t⁽⁻³⁾ˣ⁽⁻¹⁾ = t³
Therefore, the simplified expression is t³.
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A Park is shaped like a trapezoid. It has an area of 7.2 mi.² one piece is 1.2 miles on the other bases 2.4 miles long. What is the height of the trapezoid
(Can't delete my answer)
Please help it’s due today!!!!
4. RSTU is a trapezoid because the opposite sides RS and UT are parallel.
5. RSTU is not an isosceles trapezoid because the diagonals are not congruent.
How to verify that RSTU is a trapezoid?In order to verify that RSTU is a trapezoid, we would have to determine slope of the pair of opposite sides and check whether at least one pair of opposite sides are parallel;
RU ║ ST
Slope of side RU = Slope of side ST
Slope of RU = (y₂ - y₁)/(x₂ - x₁)
Slope of RU = (1 + 3)/(5 + 3)
Slope of RU = 4/8
Slope of RU = 0.5.
Slope of RS = (y₂ - y₁)/(x₂ - x₁)
Slope of RS = (-9 + 3)/(-4 + 3)
Slope of RS = -6/-1
Slope of RS = 6.
Slope of ST = (y₂ - y₁)/(x₂ - x₁)
Slope of ST = (-2 - 1)/(10 - 5)
Slope of ST = -3/5
Slope of ST = -0.6.
Slope of UT = (y₂ - y₁)/(x₂ - x₁)
Slope of UT = (-2 + 9)/(10 + 4)
Slope of UT = 7/14
Slope of UT = 0.5.
Therefore, RSTU is a trapezoid because the opposite sides RS and UT are parallel.
Question 5.
In order to determine whether RSTU is an isosceles trapezoid, we would have to determine length of the diagonals by using the distance formula and check whether they are congruent;
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Distance RT = √[(-2 + 3)² + (10 + 3)²]
Distance RT = √170 units.
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Distance US = √[(5 + 4)² + (1 + 9)²]
Distance US = √181 units.
Therefore, RSTU is not an isosceles trapezoid because the diagonals RT and US are not congruent.
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MA.7.AR.4.1
Johnny and Eleanor went to their local gas station to collect information about the cost of
fuel for compact cars. They observed both regular and premium gas purchases that day and
recorded their data in the table below.
Gallons Purchased 11.5 7.2
10
14.3 6.8
9.7
Cost
$25.23 $15.80 $21.94 $40.63 $14.92 $27.56
Part A. Is there a proportional relationship between the number of gallons of gas sold and
the cost? Explain your answer.
Part B. If the relationship is not proportional, which data value or values should be
changed to make the relationship proportional? What could explain this
difference?
a) The fourth and sixth data points have significantly different ratios.
a) Let's calculate the ratios:
For the first data point:
= 11.5 gallons / $25.23 = 0.4555 gallons per dollar
For the second data point:
= 7.2 gallons / $15.80 = 0.4557 gallons per dollar
For the third data point:
= 10 gallons / $21.94 = 0.4556 gallons per dollar
For the fourth data point:
= 14.3 gallons / $40.63 = 0.3519 gallons per dollar
For the fifth data point:
= 6.8 gallons / $14.92 = 0.4555 gallons per dollar
For the sixth data point:
= 9.7 gallons / $27.56 = 0.3517 gallons per dollar
However, the fourth and sixth data points have significantly different ratios.
b) If the relationship is not proportional, the data values that should be changed to make the relationship proportional are the fourth and sixth data points.
The difference in ratios could be explained by factors such as fluctuations in gas prices or differences in gas grades.
To establish a proportional relationship, it would be necessary to collect data where the price per gallon remains constant for all data points or to separate the data based on gas grades and analyze each grade separately.
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Can someone please answer and provide an explanation for these problems?
Answer:
details:
54)Write the midpoint formula::
Substitute
into
::
Swap the sides:
Swap the sides:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Calculate the product or quotient:
Rearrange variables to the left side of the equation:
Calculate the sum or difference:
Express solutions in ordered pairs::
(55)Write the midpoint formula::
Substitute
into
::
Swap the sides:
Swap the sides:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Calculate the product or quotient:
Rearrange variables to the left side of the equation:
Calculate the sum or difference:
Express solutions in ordered pairs::
(56)Write the midpoint formula::
Substitute
into
::
Swap the sides:
Swap the sides:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Calculate the product or quotient:
Rearrange variables to the left side of the equation:
Calculate the sum or difference:
Express solutions in ordered pairs::
(57)Write the midpoint formula::
Substitute
into
::
Swap the sides:
Swap the sides:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Calculate the product or quotient:
Rearrange variables to the left side of the equation:
Calculate the sum or difference:
Express solutions in ordered pairs::
50 Points! Multiple choice algebra question. Photo attached. Thank you!
The angle that is not a solution to the trigonometric equation sin(2θ) = 1 is given as follows:
a) 90º.
How to solve the trigonometric equation?The trigonometric equation in the context of this problem is defined as follows:
sin(2θ) = 1
The sine assumes a value of 1 at:
90º.
Hence the solution is obtained as follows:
2θ = 90º.
θ = 45º.
The equivalent angle to 45º on the third quadrant is given as follows:
180º + 45º = 225º.225 - 360 = -135º.Hence 225º and -135º are also solutions to the trigonometric equation.
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Un fabricante de bombillas gana 0,60€ por cada bombilla que sale de fábrica pero pierde 0,80€ por cada una que sale defectuosa. Un determinado día en el que fabrico 2.100 bombillas obtuvo un beneficio de 966€. ¿Cuántas bombillas de cada tipo fabricó?
The manufacturer made 1,890 non-defective light bulbs and 210 defective light bulbs.
How many defective and non-defective light bulbs was made?Let represent number of non-defective light bulbs with x
Let represent number of defective light bulbs with y.
The profit from selling non-defective light bulbs is €0.60x
The loss from selling defective light bulbs is €0.80y.
Given that the manufacturer made a profit of €966, we can form the equation:
0.60x - 0.80y = 966 ---(1)
The total number of light bulbs produced is 2,100, so we have:
x + y = 2,100 ---(2)
To solve this, we will use method of elimination.
We multiply equation (2) by 0.60.
0.60x + 0.60y = 1,260 ---(3)
Now, subtract equation (3) from equation (1):
0.60x - 0.60x - 0.80y - 0.60y = 966 - 1,260
-1.40y = -294
y = -294 / -1.40
y = 210
Substituting value of y into equation (2):
x + 210 = 2,100
x = 2,100 - 210
x = 1,890.
Translated question:
A light bulb manufacturer earns €0.60 for each bulb that leaves the factory but loses €0.80 for each one that is defective. On a certain day when he manufactured 2,100 light bulbs, he made a profit of €966. How many light bulbs of each type did he make?
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A cyclist covers 5/7 of his route and an additional 40 minutes. He has yet to cover 118 miles less than 0.75 of his route. How long is his route?
The distance that describes how long his route was is: 168 miles.
How to determine the length of the routeTo determine how long the route was, we need to first represent the entire unknown distance covered by this individual with s.
5/7 s + 40 minutes + (0.75s - 118) = s
5/7s + 75/100s + 40 - 118 = s
5/7s + 75/100s - 78 = s
5/7s + 75/100s - s = 78
325s = 54600
s = 168 miles.
So, the entire distance covered by the cyclist is 168 miles.
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The distance that describes how long his route was is: 168 miles.
How to determine the length of the route.
To determine how long the route was, we need to first represent the entire unknown distance covered by this individual with s.
5/7 s + 40 minutes + (0.75s - 118) = s
5/7s + 75/100s + 40 - 118 = s
5/7s + 75/100s - 78 = s
5/7s + 75/100s - s = 78
325s = 54600
s = 168 miles.
So, the entire distance covered by the cyclist is 168 miles.
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What can be interpreted from the scatter plot above
The thing that can be interpreted from the scatter plot above is A. There is a negative association between number of hours of daily TV viewing and math quiz scores.
How to explain the scatterplotAs per given scatter plot, If the number of hour of daily TV viewing increases then the math quiz scores are decreases.
Hence, the number of hour of daily TV viewing and math quiz scores varies inversely.
Therefore, There is a negative ath association between number of hour of daily TV viewing and math quiz scores
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What is the meaning of "possibly some ui are not free, or even do not occur, in ϕ"?
The phrase "possibly some ui are not free, or even do not occur, in ϕ" refers to a logical formula or expression ϕ that contains variables ui.
Answer to the aforementioned questionWhen it says "possibly some ui are not free," it means that not all of the variables ui in ϕ are unrestricted or unconstrained. Some of them may have certain conditions or restrictions placed on them within the context of the formula ϕ.
On the other hand, when it says "or even do not occur, in ϕ," it means that some variables ui may not appear at all in the formula ϕ. This implies that those variables are not present or relevant in the expression ϕ and do not play a role in its interpretation or evaluation.
In summary, the phrase indicates that in the formula ϕ, some variables ui may have restrictions or conditions placed on them, while others may not be present or relevant within the context of the formula.
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