d=2 and (a-1/2)2=2 hence 2a-1=2 hence a=3/2=first term and the common difference=d=2
S1, S2,... Sn can be written as:
A (first term) (first term)
A+r (r is a common difference) (r is a common difference),
A+2r
A+3r
A +(n-1)r
It is possible to calculate the sum as A +A +... +A (n times) + r +2r +3r.... +(n-1)r.
Giving n*A +r*(1+2+3....+n-1)
1+2+3….+n-1= n*(n-1)/2 =n^2 - n/2
The first N terms are added together as follows: n*A + r/2 *n2 - r*n/2
= r/2 *n^2 + (A - r/2)*n
This equals n2 - 2n for any n given the questions, so
A-r/2 = 2 AND r/2 = 1
=> r = 2 and A = 3 with r = 2
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Full question
The sum of the first n terms of an arithmetic sequence is given by Sn= n^2-2n. What are the first term and the common difference?
In an isosceles triangle ABC with AC = BC, point M is on the side BC , and point N is on the segment MC. It is known that MN=AM, and m∠BAM=m∠NAC. Find m∠MAC.
Please answer correctly in less than 10 min
The measure of angle MAC is 90°.
What does the isosceles triangle mean?
When two of a triangle's sides are congruent, or the same length, the triangle is said to be isosceles. This means that if you draw a line through the middle of an isosceles triangle, it will split in half, with the two parts having equal-length sides.
The angles on either side of the congruent side of an isosceles triangle are also congruent. As a result, in the ABC triangle, mBAM = mCAB = mNAC.
Angle MAC is congruent to itself since angles BAM and NAC are complementary to it and are also congruent with each other.
Therefore, m∠MAC = m∠MAC = 180 - m∠BAM = 180 - m∠NAC.
Since m∠BAM = m∠NAC, we can conclude that
m∠MAC = 180 - m∠NAC = 90°
Hence, the measure of angle MAC is 90°.
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Given measure angle ABC = 120° and measure angle CBD = 78°. According to the angle addition postulate, what he the measure of angle ABD, that contains BC?
According to the angle addition postulate, the measure of ∠ABD, that contains BC is 162°.
We have to determine according to the angle addition postulate, the measure of ∠ABD, that contains BC.
Measure of ∠ABC = 120° and measure of ∠CBD = 78°.
As we know that the sum of all angle is 360°.
So; ∠ABC + ∠CBD + ∠ABD = 360°
Now putting the value
120° + 78° + ∠ABD = 360°
Simplify
198° + ∠ABD = 360°
Subtract 198° on both side, we get
∠ABD = 162°
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The diagram of the question is:
The sides of a triangle are measured at a = 4, b = 5 and c = 6. What is the length of Median A?
A pipe takes 6 minutes to fill 3.4 of a water tank how long would it take to fill the entire tank?
x = 20.4 it take to fill the entire tank.
How to calculate how long would it take to fill the entire tank?Calculate a tank's total capacity and the volume that is filled in gallons and litres for tanks like water and oil storage. assumes the tank's interior measurements.
You can enter metric dimensions in metres (m) or centimetres, or U.S. dimensions in feet (ft) or inches (in) (cm). Results are given in fluid US gallons, fluid Imperial (UK) gallons, fluid cubic feet (ft3), fluid metric litres (fl), and fluid cubic metres (m3).
x = total volume of the entire tank.
We can suggest the following equation:
6 minutes
6 minutes to fill 3.4
x=6*3.4=20.4
x=20.4
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whats the answer for 45,688 -5.65
Answer:
Step-by-step explanation:
Help, it's something that has to do with Slope-intercept form
Answer:
y = 1/2x + 7/6
Step-by-step explanation:
Where does 19/8 go on a graph
The fraction 19/8 on the graph is located between the integers 2 and 3, exactly at the decimal 2.375.
How to convert a fraction to decimal?A fraction is converted to decimal applying a proportion, which is the division of the numerator of the fraction by the denominator of the fraction.
The fraction in this problem is given as follows:
19/8.
For which:
The numerator is of 19.The denominator is of 8.Hence the equivalent decimal is given as follows:
19/8 = 16/8 + 3/8 = 2 + 0.375 = 2.375.
Which is between 2 and 3 on the number line.
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What is the median of this data set?
{15, 14, 6, 10, 7, 15, 7, 8}
Enter vour answer in the box
The median of the dataset is 9.
What is the median ?
Median is a measure of central tendency that is used to determine the number that is at the center of a dataset. The dataset has to be arranged in either ascending or descending order.
The data set arranged in ascending order : 6, 7, 7, 8, 10, 14, 15, 15
Median = (n + 1) / 2 = (8 + 1) / 2 = 9/2 = 4.5th number = (10 + 8) / 2 = 18/2 = 9
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Select all the equations where x=-2 is a solution
check all that apply
A. 4x=4+2x
B. 2(x+5)=x+8
C. 5+3x=-1
D. 3x-5=1
E. 19=2(x-6)+3
The equations that have x = -2 as their solution are B and C
How to solve the equationHow to find the solution x = - 2
A. 4x=4+2x
4x - 2x = 4
2x = 4
x = 4 / 2
x = 2
B 2(x+5)=x+8
2x + 10 = x + 8
2x - x = 8 - 10
x = -2
C. 5+3x=-1
5 + 1 = - 3x
6 = -3x
x = -2
D 3x-5=1
3x = 5 + 1
3x = 6
x = 2
E 19=2(x-6)+3
19 = 2x - 12 + 3
19 = 2x - 9
19 + 9 = 2x
28 = 2x
x = 28/2
x = 14
The equations where x=-2 is a solution are 5+3x=-1 , 2(x+5)=x+8
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The function f(x) = 12,000(1.036)x, represented in the table, shows the population of a certain town where x is the number of years after 2008.
Year 2008 2011 2014 2017 2020
Population 12,000 13,343 14,836 16,498 18,344
Find the average rate of change from 2014 to 2017 and interpret its meaning in terms of the context.
−831; represents the average depreciation in the population from 2014 to 2017
831; represents the average growth in the population from 2014 to 2017
−554; represents the average depreciation in the population from 2014 to 2017
554; represents the average growth in the population from 2014 to 2017
The average rate of change from 2014 to 2017 is 554.
What is Average rate of Change?It is the average amount by which the function changed per unit throughout that time period. It is calculated using the slope of the line linking the interval's ends on the graph of the function.
Given:
f(x) = 12,000[tex](1.036)^x[/tex]
Year 2008 2011 2014 2017 2020
Population 12,000 13,343 14,836 16,498 18,344
Now, the average rate of change from 2014 to 2017
= f(2017)- f(2014)/(2017- 2014)
= 12,000[tex](1.036)^{2017[/tex] - 12,000[tex](1.036)^{2014[/tex]/3
= 1662/ 3
= 554.
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Five years ago, a computer that worked half as fast as a current computer cost twice as much money. The current computer's speed to price ratio is what percent of the older computer's speed to price ratio? I'm confused on what the ratio is
The current computer's speed to price ratio is 400% of the older computer's speed to price ratio.
How to obtain the speed to price ratios?
The speed to price ratios, along with the percentage, is obtained applying the proportions in the context of this problem.
Five years ago, the ratio was given as follows:
0.5x/2y = 0.25x/y.
The current ratio is given as follows:
x/y.
The current ratio is 4 times greater than the ratio of five years ago, hence it's a percentage of 400% the old ratio.
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Fred bought a soft drink for 2 dollars and 7 candy bars. He spent a total of 16 dollars. How much did each candy bar cost ?
Answer:
$2 for each candy bar
Step-by-step explanation:
16 - 2 = 14
14 / 7 = 2
Answer:
$2
Step-by-step explanation:
16-2=14
14/7=2
he spent $14 total on candy bars, divide that by 7, and each candy bar costs $2
at what intercepts is the function f(x)=-1/4x^3 constant ?
The function f(x) = 1/4x³ is constant when the intercept is 0
What are polynomial graph?Polynomials are sums of terms of the form k⋅xⁿ, where k is any number and n is a positive integer.
the graph of a polynomial function is a smooth continuous curve.
The y-intercept is the point where the graph crosses the y-axis and can be found by substituting x = 0. The x-intercepts are points where the graph crosses the x-axis and can be found by making the function equal to zero and solving for x.
therefore f(0) = 1/4 0³ = 0
therefore the y- intercept of the function is 0
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Question 3 of 5
Tina's height is 6 inches less than her brother's height. Her brother's height is unknown.
Which expression shows Tina's height? Use × for her brother's height.
O A. x-6
O B. *
• C. 6x
• D. x+6
If EF = 2x, FG = x + 1, and EG = 4, what is EF?
Given that EF = 2x, FG = x + 1, and EG = 4, we can use the information provided to find the value of EF.
Since EG = 4, we know that EF + FG = EG.
so, 2x + (x + 1) = 4
Solving for x, we get:
3x + 1 = 4
3x = 3
x = 1
Now we can substitute this value back into the equation for EF:
EF = 2x
EF = 2(1)
EF = 2
So, EF = 2
It is not possible to find the value of FG if the information provided is not correct and it's missing a part of the equation, Therefore I'm unable to find the value of FG.
Given BE ≅ BD and AD ≅ CE, prove triangle ABE ≅ triangle CBD
Answer: To prove that triangle ABE is congruent to triangle CBD, we will use the SAS (Side-Angle-Side) Congruence Theorem.
This theorem states that if two triangles have two sides and the included angle congruent, then the triangles are congruent.
Given:
BE ≅ BD (congruent sides)
AD ≅ CE (congruent sides)
We have congruent sides AB and BC, and congruent angles A and C (included angles)
By SAS congruence theorem, we can prove that:
Triangle ABE ≅ triangle CBD
Therefore, triangle ABE is congruent to triangle CBD
Alternatively, if we use the ASA (Angle-Side-Angle) congruence theorem, where we have congruent angles and congruent sides between those angles, we can also prove that triangle ABE ≅ triangle CBD
Step-by-step explanation:
4 in
10 in
25 in
4 in 4 in
25 in
10 in
10 in
4 in
What is the surface area of the solid?
OA. 780 square inches
OB. 390 square inches
O c. 700 square inches
OD. 1000 square inches
Answer:
B
because if yiy multiple all those you get somewhere around 390 and then you round the number it is 390
Matthew and David have $689.45. If David has $346.90, how much does Matthew have? Write and solve an additional equation to find how much money belongs to Matthew.
Answer: 342.55
Step-by-step explanation:
689.45 - 346.90 = 342.55
or
346.90 + 342.55 = 689.45
) Wanda, a caterer, is investing some money in equipment and employees
to help grow her business. Recently she spent $89 on equipment and hired a
server who makes $16 per hour. Wanda is hoping to make up these expense
at the next job that is scheduled, which pays a base of $90 in addition to $15
per hour that the server works. In theory, this event could pay enough to
cancel out Wanda's expenditures. How much would the job pay? How long
would the job have to be?
The total payment will be $90 + $15x and the job would have to be 1/15 hour long for the payment to cancel out Wanda's expenditures.
What is Fraction?
Fraction is a mathematical expression that represents a part of a whole. It is written as a ratio of two numbers, where the first number is the numerator and the second number is the denominator. For example, 1/2 is a fraction that represents one-half of a whole.
The total cost of equipment and employee wages for a business is the sum of the cost of the equipment and the cost of the employee's wages, which is calculated by multiplying the employee's hourly wage by the number of hours worked.
To determine how much the job would pay, we need to calculate the total cost of the equipment and the server's wages.
The cost of the equipment is $89 and the server's wage is $16 per hour.
To calculate the total cost of the server's wages, we need to know how long the job is. Let x be the number of hours the server works.
Therefore, the total cost of the server's wages is $16x
The job pays a base of $90 in addition to $15 per hour that the server works. So the total payment is
90 + 15x
To cancel out Wanda's expenditures, The total payment should be equal to the total cost of the equipment and the server's wages. So, we can set up the equation:
$90 + $15x = $89 + $16x
To solve for x, we can subtract $89 from both sides:
$15x = $1
Finally, we can divide both sides by 15
x= 1/15
So, the job would have to be 1/15 hours long for the payment to cancel out Wanda's expenditures.
It should be noted that this answer is not realistic, since time cannot be expressed in fractions of hours. Also, the job payment would be enough to cover the costs but not to generate any profit for Wanda.
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Create a diagram based on the given symbolic statement. Be sure enough information is provided in the diagram to validate each statement.
△FUN≅△CAR
According to the diagram,
The two triangles are right-angle triangles.
ΔFUN and ΔCAR
where angle CAR and angle FUN are 90° each.
According to RHS congruency
NU = CA (each 3 cm)
FUN = CAR (each 90°)
FN = CR (each 5cm)
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Find the value of y
please and thank you
The value of y of the given rhombus is; y = 7
How to find the length of the rhombus?The parallel sides of a rhombus or even parallelogram are always equal to each other and as a result in this case, the parallel sides are equal. Thus;
3y - 12 = 5y - 26
Rearrange to get;
5y - 3y = 26 - 12
2y = 14
y = 14/2
y = 7
Thus;
Length of side of rhombus = 3(7) - 12
= 21 - 12
= 9
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3x+9=2x+5 what is the answer
Answer:
-4
Step-by-step explanation:
3x + 9 = 2x + 5
Collect like terms
3x - 2x = 5 - 9
x = -4
________ is measured in square units because it has two dimensions: length and width.
A. Area
B. Height
C. Perimeter
D. Volume
The quantity that is measured in square units is area ( option C )
What is area?Area is defined as the total space taken up by a flat (2-D) surface or shape of an object. Take a pencil and draw a square on a piece of paper. It is a 2-D figure. The space the shape takes up on the paper is called its Area.
The area of a rectangle is given as Length × width . The area of a triangle is 1/2 bh.
Height has only one dimension. it is measured in meters. Perimeter has only one dimension, it is also measure in meters. Volume has three dimensions i.e l × b× h = m³
Therefore Out of the options only area has two dimension i.e l× w= m× m = m²
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you measure the period of a mass oscillating on a vertical spring ten times as follows: period (s): 1.88, 0.97, 1.43, 1.26, 1.61, 1.19, 1.17, 1.28, 1.25, 1.48 what are the mean and (sample) standard deviation?
The mean period of the oscillation is 1.257 s and the standard deviation is 0.123 s.
To find the mean and standard deviation of the data, we can use the following formulas:
The mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set. The median is the middle value when a data set is ordered from least to greatest
Mean = (sum of data points) / (number of data points)Standard deviation = [tex]\frac{\sqrt{((sum of (data point - mean)^{2}) } \\}{number of data points - 1}[/tex]
First, let's find the mean:
Mean = (1.88 + 0.97 + 1.43 + 1.26 + 1.61 + 1.19 + 1.17 + 1.28 + 1.25 + 1.48) / 10
Mean = [tex]\frac{12.57}{10}[/tex]
Mean = 1.257
The mean is 1.257.
Next, let's find the standard deviation:
Standard deviation = ([tex]\sqrt{(((1.88 - 1.257)^{2}+(0.97 - 1.257)^{2} (1.43 - 1.257)^{2}+(1.26 - 1.257)^{2}+(1.61 - 1.257)^{2}}[/tex][tex]+\sqrt{+(1.17 - 1.257)^{2}+(1.28 - 1.257)^{2}+(1.48 - 1.257)^{2}}[/tex])/9
Standard deviation =[tex]\sqrt{\frac{0.134}{9} }[/tex]
Standard deviation = [tex]\sqrt{0.015}[/tex]
Standard deviation = 0.123
Therefore, the mean period of the oscillation is 1.257 s and the standard deviation is 0.123 s (sample standard deviation).
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a polling agency is investigating the voter support for a ballot measure in an upcoming city election. the agency will select a random sample of 500 voters from one region, region a, of the city. assume that the population proportion of voters who would support the ballot measure in region a is 0.47.
According to the normal distribution and the central limit theorem the ballot measure is greater than 0.50.
The term called normal distribution is defined as a symmetrical plot of data around its mean value, where the width of the curve is defined by the standard deviation.
Here we know that t a random sample of 500 voters from one region, region a, of the city and the ballot measure in region a is 0.47.
Here we have to find the mean and the standard deviation are calculated as,
=> mean = 0.47
=> Standard deviation = √0.47(1-0.47)/500
When we simplify this then we get,
=> Standard deviation = 0.0223
By using the Central Limit Theorem the probability is calculated as,
=> Z = (0.5 - 0.47) / 0.0223
=> Z = 1.34
When we use the z table, then we get the p - value as 0.9099
Then the probability is calculated as,
=> P = 1 - 0.9099
=> P = 0.0901
Hence 9.01% probability that the proportion of voters in the sample of Region A who support the ballot measure is greater than 0.50.
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URGENT!
Find the point of intersection of the diagonals of the rhombus with vertices:
(-1,2), (3,4), (5,8), and (1,6)
The point of intersection of the diagonals of the rhombus with vertices is (3, -5)
What is a function?Function is a type of relation, or rule, that maps one input to specific single output.
When a relation is given in the form of ordered pairs, the first element of each ordered pair is the member of domain set and second element is the member of range set.
Given the rhombus with vertices:
(-1,2), (3,4), (5,8), and (1,6)
The domain is:
{1,3,5,1}
Rhombus is a parallelogram in which diagonals bisect each other, we can see there is repetition in domain i.e. 1 is twice in the set, the given relation is not a function.
Hence, the point of intersection is (3, -5)
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explain why the exponents can not be added in the product 12 to the power of 3 then * 11 to the power of three
Answer:
In the product 12 to the power of 3 * 11 to the power of 3, the exponents (3 and 3) indicate that 12 is multiplied by itself 3 times, and 11 is multiplied by itself 3 times.
When two or more numbers with the same base are multiplied together, the exponents can be added, because it's the same as multiplying the base numbers together. For example, (12^3) * (12^2) = 12^(3+2) = 12^5
However, in this case, the bases of 12 and 11 are different, and therefore, the exponents cannot be added. The product of 12 to the power of 3 * 11 to the power of 3 is different from (1211) to the power of 3, the result is (1211)^3= 1332^3
Stacy sells art prints for $12 each. Her expenses are $2. 50 per print, plus $38 for equipment. How many prints must she sell for her revenue to equal her expenses?
Answer:
Stacy needs to sell at least 23 prints in order for her revenue to equal her expenses.
Explanation: To calculate the total expenses, we need to add the fixed cost of equipment ($38) to the variable cost of each print ($2.5). So the total expenses per print is $2.5 + $38 = $40.50. To calculate the number of prints needed to break even, we divide the total expenses by the revenue per print. 40.50 / 12 = 3.375 which is approximately 3.4 and if we round it up to 4 ,then we will get 4*12 = 48 and this is less than the fixed cost of 38. So, we need to increase the number of print to 23 in order to her revenue to equal her expenses.
Step-by-step explanation:
Jackie made the same
amount of bracelets for
8 days. Write an expression
for the number of bracelets
made.
Jackie made 12 bracelets each day and the equation used to represent this is 8x = 96. The solution has been obtained using the concept of simplification.
What is simplification?
In mathematics, the operation and interpretation of a function to make it simple or easier to grasp is known as simplifying, and the process is known as simplification.
We are given that Jackie made bracelets for 8 days and when he was done he had 96 bracelets with him.
Let 'x' be the number of bracelets made by Jackie each day
So, we get
⇒8x = 96
⇒x = 96/8
⇒x = 12
Hence, Jackie made 12 bracelets each day and the equation used to represent this is 8x = 96.
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Question: Jackie made bracelets for 8 days. When he was done he had 96 bracelets Write equation to express how many bracelets Jackie made each day.
Darren is placing shipping boxes in a storage unit with a floor area of x* + 5x° + x7 _ 20x - 14 square units. Each box has a volume of x3 + 10x2 + 29x + 20 cubic units and can hold a stack of items with a height of x + 5 units.
a. How much floor space will each box cover?
b. What is the maximum number of boxes Darren can place on the floor of the storage unit?
c. Assume Darren places the maximum number of boxes on the floor of the storage unit, with no overlap. How much of the floor space is not covered by a box?
Answer:
Step-by-step explanation:
a. To find out how much floor space each box will cover, we need to multiply the volume of the box by the height of the stack of items. The volume of the box is x3 + 10x2 + 29x + 20 cubic units and the height of the stack of items is x + 5 units. So the floor space covered by each box is (x3 + 10x2 + 29x + 20) * (x + 5) square units.b. To find the maximum number of boxes Darren can place on the floor of the storage unit, we need to divide the total floor area of the storage unit by the floor space covered by each box. The total floor area of the storage unit is x* + 5x° + x7 _ 20x - 14 square units and the floor space covered by each box is (x3 + 10x2 + 29x + 20) * (x + 5) square units, then we can divide the total floor area by the floor space covered by each boxx* + 5x° + x7 _ 20x - 14 / (x3 + 10x2 + 29x + 20) * (x + 5) = maximum number of boxesc. To find out how much of the floor space is not covered by a box, we can subtract the floor space covered by the maximum number of boxes from the total floor area of the storage unit.Total floor area - floor space covered by the maximum number of boxes = remaining floor space.Please note that this is a polynomial equation and the solution will be different depending on the value of x.
a) Floor space of each box = ([tex]x^3 + 10x^2 - 29x + 20[/tex]) / (x + 5)
b) Maximum number of boxes = ([tex]x^4 + 5x^3 + x^2 - 20x - 14[/tex]) / (([tex]x^3 + 10x^2 - 29x + 20[/tex]) / (x + 5))
c) Uncovered floor space = [tex]x^4 + 5x^3 + x^2 - 20x - 14 - ((x^3 + 10x^2 - 29x + 20) / (x + 5) * Maximum number of boxes)[/tex]
We have a storage unit with a given floor area and boxes with specified volumes and heights.
We'll determine the floor space each box covers, the maximum number of boxes Darren can place on the floor, and how much floor space is left uncovered when he places the maximum number of boxes without overlap.
a) Floor space covered by each box:
To find the floor space covered by each box, we divide the volume of the box by its height. This gives us the area each box covers on the floor.
The floor space of each box can be calculated by dividing its volume by its height.
The volume of each box is given by the expression [tex]x^3 + 10x^2 - 29x + 20[/tex], and its height is x + 5. Therefore, the floor space covered by each box can be represented as:
Floor space of each box = Volume of the box / Height of the box
Floor space of each box = [tex](x^3 + 10x^2 - 29x + 20) / (x + 5)[/tex]
b) Maximum number of boxes Darren can place:
To find the maximum number of boxes Darren can place on the floor, we need to divide the total floor area by the floor space covered by each box.
The maximum number of boxes Darren can place is determined by dividing the total floor area, given by [tex]x^4 + 5x^3 + x^2 - 20x - 14[/tex], by the floor space covered by each box. We already calculated the floor space of each box as [tex](x^3 + 10x^2 - 29x + 20) / (x + 5)[/tex]. Now, we'll perform the division:
Maximum number of boxes = Total floor area / Floor space of each box
Maximum number of boxes = [tex](x^4 + 5x^3 + x^2 - 20x - 14) / ((x^3 + 10x^2 - 29x + 20) / (x + 5))[/tex]
c) Uncovered floor space with maximum number of boxes:
To find the uncovered floor space when Darren places the maximum number of boxes without overlap, we subtract the total floor area covered by the boxes from the total floor area of the storage unit.
The floor space not covered by boxes can be determined by subtracting the total floor area covered by the boxes from the total floor area of the storage unit.
We already know the total floor area of the storage unit is [tex]x^4 + 5x^3 + x^2 - 20x - 14[/tex], and we calculated the floor space covered by each box in part (a). The maximum number of boxes Darren can place was determined in part (b). Now, we'll perform the subtraction:
Uncovered floor space = Total floor area of storage unit - (Floor space of each box × Maximum number of boxes)
Uncovered floor space =[tex]x^4 + 5x^3 + x^2 - 20x - 14 - ((x^3 + 10x^2 - 29x + 20) / (x + 5) * Maximum number of boxes)[/tex]
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Complete Question
Darren is placing shipping boxes in a storage unit with a floor area of
[tex]x^4+5x^3+x^2-20x-14[/tex] square units. Each box has a volume of [tex]x^3+10x^2-29x+20[/tex].
cubic units and can hold a stack of items with a height of x + 5 units.
a) How much floor space will each box cover?
b) What is the maximum number of boxes Darren can place on the floor of the storage unit?
c) Assume Darren places the maximum number of boxes on the floor of the storage unit, with no overlap. How much of the floor space is not covered by a box?