The exact value of x (in ft) so that the greatest possible amount of light is admitted is 8.96 ft
According to the question,
A norman window has the shape of a rectangle surmounted by a semicircle.
Let The width of rectangle be x and height be y
It is given that the diameter of the semicircle is equal to the width of the rectangle
Therefore , Diameter of circle = x
Radius = x/2
Perimeter of window = 32 = Perimeter of rectangle + Perimeter of semi-circle
=> 32 = Width + 2height + πr
=> x + 2y + πx/2 = 32
=> 2x + 4y + πx = 64
Solving for y,
=> 4y = 64 - 2x - πx
=> y = 64 - 2x - πx/ 4 ----------(1)
Area of the window = Area of rectangle + area of semi-circle
=> A = Width × hiegth + πx²/8
=> A = xy + πx²/8
Substituting the value of y from equation (1)
=> A = x ( 64 - 2x - πx/ 4 ) + πx²/8
[tex]A = x ( \frac{64 - 2x -\pi x}{ 4} ) + \frac{\pi x^2}{8}\\A = ( \frac{ 64x - x^2 ( 2 + \pi)}{4 } ) + \frac{\pi x^2}{8}[/tex]
Differentiating A w.r.t x
=> [tex]\frac{dA}{dx} = (\frac{64 - 2x(2 + \pi)}{4} ) + \frac{\pi x}{4}[/tex]
Now , To find a maximum area
dA/dx = 0
=> [tex]\frac{dA}{dx} = (\frac{64 - 2x(2 + \pi)}{4} ) + \frac{\pi x}{4} = 0[/tex]
Solving for x ,
64 - 4x - 2π + πx = 0
=> x = 64 / 4 + π ------(2)
Differentiating A again w.r.t x
=> d²A / dx² = -2(2 + π)/4 + π/4
=> -π+4 / 4 which is less than zero
Therefore,
Area is maximum when x = 64 / 4 + π
The exact value of x (in ft) so that the greatest possible amount of light is admitted is 8.96 ft
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a box contains ten sealed envelopes numbered 1, . . . , 10. the first six contain no money, the next two each contains $5, and there is a $10 bill in each of the last two. a sample of size 3 is selected with replacement (so we have a random sample), and you get the largest amount in any of the envelopes selected. if x1, x2, and x3 denote the amounts in the selected envelopes, the statistic of interest is m
The probability distribution for the statistic is ; for m = 0 , P(0) = 0.216 , for m = 5 ,P(5) = 0.296 and for m = 10 , P(10) = 0.488 .
In the question ,
it is given that ,
the box has 10 sealed envelopes , that contains number from 1,2,...,10 .
the first six envelopes has no money .
next two envelopes has $5
and the last two contains $10 bill .
if x₁, x₂, and x₃ denote the amounts in the selected envelopes,
the statistic of interest is m = maximum of X₁ , X₂ and X₃ .
we need to find probability distribution for the statistic m,
So , P(m = 0) = P( all three envelope contains no money) = (6/10)³ = 0.216
and P(m = 5) = P( all from 8 envelope which contains at most $5) - P(all three envelope with 0 money)
= (8/10)³ - 0.216
= 0.296
and P(m = 10) = 1 - (8/10)³ = 0.488
Therefore , the required probability is P(0) = 0.216 , P(5) = 0.296 and P(10) = 0.488 .
The given question is incomplete , the complete question is
A box contains ten sealed envelopes numbered 1, . . . , 10. the first six contain no money, the next two each contains $5, and there is a $10 bill in each of the last two. a sample of size 3 is selected with replacement (so we have a random sample), and you get the largest amount in any of the envelopes selected. if x₁, x₂, and x₃ denote the amounts in the selected envelopes, the statistic of interest is M = the maximum of X₁ ; X₂ , and X₃ . Obtain the probability distribution of this statistic ?
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Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 27x^3, y = 0, x = 1; about x = 2
The volume V of the solid is obtained by rotating the region bounded by the given curves about the specified line. y = 27x^3, y = 0, x = 1; about x = 2 is 81π/5
Shell method:
The shell method is an integration for calculating the volume of a solid of revolution when integrating along an axis perpendicular to the axis of revolution. This is in contrast to disc integration which integrates along the axis parallel to the axis of revolution.
V=2π∫01(2-x). 27x³dx
= 2π.27([tex]x^{4}[/tex]/2 - [tex]x^{5}[/tex]/5)01
=54π(1/2-1/5)
= 81π/5
Therefore The volume V of the solid is obtained by rotating the region bounded by the given curves about the specified line. y = 27x^3, y = 0, x = 1; about x = 2 is 81π/5
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-1/2h(2h+3)=0 answer
Step-by-step explanation:
-1/2h(2h+3) = 0
Seperate the equation:
-1/2 = 0
h = 0
(2h+3) = 0
Subtract 3 from both sides:
2h+3 = 0
2h+3-3 = 0
Simply:
2h = -3
Final Answer:
h = 0
h = -3/2
60−40y distributive property
Answer:
20(3-2y)
Step-by-step explanation:
60−40y =
10(6-4y) = ==> both 60 and 40 are multiples of 10
2(10(3-2y)) = ==> both 6 and 4 are factors of 2
2*10(3-2y) = ==> simplify
20(3-2y)
the sugar sweet company delivers sugar to its customers. let be the total cost to transport the sugar (in dollars). let be the amount of sugar transported (in tons). the company can transport up to tons of sugar. suppose that gives as a function of . identify the correct description of the values in both the domain and range of the function. then, for each, choose the most appropriate set of values.
The domain of the function is: [0,∞), the range of the function is: [3500,∞)
What does domain and range mean?Domain and Range: The set which contains all the first elements of all the ordered pairs of relation R is known as the domain of the relation. The set which contains all the second elements, on the other hand, is known as the range of the relation.
The potential input and output values of a function are its domain and range, respectively.
The formula for the function is C=130S+3500.
the region
This is a list of the function's potential S values.
Due to the fact that S refers to a physical quantity, it cannot be less than 0. (i.e. tons of sugar)
The domain of the function is [0,∞)
Since S can have any value larger than 0.
The variety
This represents the function's potential C values.
The formula for the function is C=130S+3500.
Assuming S = 0, the following is true: C=130*0+3500=3500.
C must be more than 3500.
Any number higher than 3500 can be used as the value of C.
Consequently, the function's range is [3500,∞)
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Suppose you have selected a random sample of n = 7 measurements from a normal distribution. Compare the standard normal z values with the corresponding t values if you were forming the following confidence intervals. (a) 95% confidence interval N (b) 80% confidence interval 23 (c) 90% confidence interval 2 = t =
(a) At 95% confidence interval z- value is 1.96 and t-value is 2.447.
(b) At 80% confidence interval z- value is 1.282 and t-value is 1.440.
(c) At 90% confidence interval z- value is 1.645 and t-value is 1.943.
Given that,
Let's say you randomly choose n=7 measurements from a normal distribution. If you were constructing the following confidence intervals, compare the standard normal z values with the appropriate t values.
We have to find
(a) At 95% confidence interval what is z- value and t-value.
(b) At 80% confidence interval what is z- value and t-value.
(c) At 90% confidence interval what is z- value and t-value.
We know that,
Sample size = n = 7
Degrees of freedom = df = n - 1 = 7 - 1 = 6
(a) At 95% confidence level
α = 1 - 95%
α = 1 - 0.95 =0.05
α/2 = 0.025
Zα/2 = Z0.025 = 1.96
z = 1.96
At 95% confidence level
α= 1 - 95%
α =1 - 0.95 =0.05
α/2 = 0.025
tα/2,df = t0.025,6 = 2.447
t = 2.447
(b) At 80% confidence level
α = 1 - 80%
α = 1 - 0.80 =0.20
α /2 = 0.10
Zα /2 = Z0.10 = 1.282
z = 1.282
At 80% confidence level
α = 1 - 80%
α =1 - 0.80 =0.20
α /2 = 0.10
tα /2,df = t0.10,6 = 1.440
t = 1.440
(c) At 90% confidence level
α = 1 - 90%
α = 1 - 0.90 =0.10
α /2 = 0.05
Zα /2 = Z0.05 = 1.645
z = 1.645
At 90% confidence level
α = 1 - 90%
α =1 - 0.90 =0.10
α /2 = 0.05
tα /2,df = t0.05,6 = 1.943
t = 1.943
Therefore,
(a) At 95% confidence interval z- value is 1.96 and t-value is 2.447.
(b) At 80% confidence interval z- value is 1.282 and t-value is 1.440.
(c) At 90% confidence interval z- value is 1.645 and t-value is 1.943.
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Solve for x.
3/7x +1/4x =38
Answer:
x = 56
Step-by-step explanation:
[tex]\frac{3}{7}[/tex] x + [tex]\frac{1}{4}[/tex] x = 38 The common denominator would be 28
[tex]\frac{12}{28}[/tex] x + [tex]\frac{7}{28}[/tex] x =38
[tex]\frac{19}{28}[/tex] x = 38 Multiply both sides by [tex]\frac{28}{19}[/tex]
[tex](\frac{28}{19})[/tex][tex]\frac{19}{28}[/tex] x = [tex](\frac{28}{19})[/tex][tex]\frac{38}{1}[/tex] Cross cancel the 19 and 38.
x = [tex](\frac{28}{1})[/tex][tex]\frac{2}{1}[/tex]
x = 56
e. 2|x-4|-3y³ when x=3 and y=-2
Answer:
26
Step-by-step explanation:
To find the value of the expression 2|x-4|-3y³ when x=3 and y=-2, we need to substitute the values of x and y into the expression and then simplify. The absolute value part of the expression, |x-4|, is the distance between the number x and 4 on the number line. When x=3, this distance is |3-4|=1. So the expression becomes 2*1-3(-2)³ = 2-3(-8) = 2+24 = 26. Therefore, when x=3 and y=-2, the value of the expression 2|x-4|-3y³ is 26.
Find the cosecant. Someone help me asap pls!
Answer:
sqrt2
Step-by-step explanation:
csc (x)= 1/sin(x)
sin(x)= (sqrt22)/(sqrt22*sqrt2)= 1/sqrt2
1/(1/sqrt2)=sqrt2/1=sqrt2
can anyone please help me with this math question?
Answer:
a) (7-2x) +3 = 10 - 2X
b) x = y + 3 <=> y = x-3
c) (x-3).-2 +7 = -2x +6 +7 => -10 + 13 = 3
In the Two Period Consumer model, the Net Effect from a decrease in the interest rate for a net borrower is:
Group of answer choices
C1↓; C2?; Savings ↑
C1↑; C2?; Savings ↓
C1↑; C2↓; Savings ↓
C1?; C2↓; Savings ?
C1?; C2↑; Savings ?
In the two period consumer model , the net effect from a decrease in the interest rate for a net borrower is increase in present consumption (C1 increases) , future consumption decreases (C2 decreases) and savings decreases.
What is two period consumer model?
The consumer lives for two cycles before passing away. Therefore, there is no use in saving during the second time period. Rearranging, our present disposable income minus our current consumption is what we call savings.By deciding not to spend their entire income during the present period, consumers conserve money. They borrow money from the credit market to cover the difference between their income and current expenditure, boosting their future period income by the amount they saved plus interest.A useful simplification, in fact. Maximum current expenditure exceeds maximum current income.Hence, the net effect from a decrease in the interest rate for a net borrower is increase in present consumption (C1 increases) , future consumption decreases (C2 decreases) and savings decreases.
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Graph the system of inequalities.
y>4 x-3
3 y-x≤9
Use the graphing tool on the right to graph the system of inequalities.
The graph of the inequality y > 4x - 3 , 3y - x ≤ 9 . is shown below.
In the question ,
the inequalities are given ,
we have to graph given inequalities ,
the inequalities are y > 4x - 3
3y - x ≤ 9 .
in the first inequality , y > 4x - 3
we put x = 0 , y is = -3 .
we put y = 0 , x is = 3/4 .
So ,the points for the line will be (0,-3) and (3/4,0) .
since the inequality does not have equal to sign , the line will be a dotted line .
in the second inequality ,3y - x ≤ 9 .
we put x = 0 , y is = 3 .
we put y = 0 , x is = -9 .
the points for the line will be (0,3) and (-9,0).
Since the inequality has equal to sign . Thus, line will be solid line .
Therefore , the for the given inequality is shown below.
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Which is the equation of line b
The equation of line b that passes through the point (-3,2) is y=-1/4x+5/4
How do you find a line equation?to get the equation of a line given a point and its slope.
Recognize the slope.
Decide on the point.
In the point-slope form, y y 1 = m (x x 1), substitute the values. y − y 1 = m (x − x 1) .
Slope-intercept form should be used to write the equation.
What are the 3 different equations of lines?There are three major forms of linear equations: point-slope form, standard form, and slope-intercept form.
In the above question we see that,
The line b passes through the point (-3,2)
the equation of line should satisfy this point
In the given option the only line that satisfy the point is
option C
y=-1/4x+5/4
hence option C ,y=-1/4x+5/4 is the answer
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What grade does she need on the 4th test to have an average of 80% on all 4 tests?
A. 77 %
B. 80 %
c. 85 %
D. 89%
E. 100%
The grade needed by Kim on the fourth test to have the average of 80% is 89% , the correct option is (d) .
In the question ,
it is given that ,
the scores that Kim got in the first three tests are 82% , 75% , 74% .
let the grade required by Kim to score average of 80% be = x% .
Since Kim needs to score average of 80% in all the four tests ,
that means ,
(82 + 75 + 74 + x)/4 = 80
After Simplifying further further ,
we get ,
(231 + x)/4 = 80
231 + x = 80 × 4
231 + x = 320
x = 89% .
Therefore , Kim needs to score 89% in the fourth test .
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Determine whether the data come from a normally distributed population. Choose the correct answer below. A. The distribution is not normal. The points are not reasonably close to a straight line. B. The distribution is normal. The points show a systematic pattern that is not a straight-line pattern. C. The distribution is not normal. The points show a systematic pattern that is not a straight-line pattern. D. The distribution is normal.
The distribution is not normal. The points show a systematic pattern that is not a straight-line pattern curved over mean.
A normal distribution is a type of data distribution in which the data points form a symmetric, bell-shaped curve around the mean.
The data points in the example provided show a systematic pattern that is not a straight-line pattern, indicating that the data does not come from a normally distributed population.
This is because the points are not reasonably close to a straight line, which is a characteristic of a normal distribution.
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A stage manager is trying to seat important guests in the front row of a theater. She would like to seat a diplomat in the first seat, a singer in the second seat, and a movie director in the third seat. If there are 3 diplomats, 2 singers, anf 2 directors attending the show, how many different front row plans are possible? (The workbook got the answer of 288) How?
Answer:
12 row plans
Step-by-step explanation:
the first seat has two possible options , the two directors. The second seat has another two options. With each director comes 2 options for the second seat so 2×2=4 possible options for the first and second seat . The 3rd seat has three options, that means for each combination of the 1st and 2nd seat three possible options 2×3=12 combinations
Ben and Bob made a snowman been spent two hours more than double the time Bob spent
Determine how many times the innermost loop will be iterated when the algorithm segment is implemented and run. (Assume that m and n are positive integers.)
for j := 1 to m
for k := 1 to n
[Statements in body of inner loop.
None contain branching statements that
lead outside the loop.]
next k
next j
Discrete mathematics
The inner loop iterates 'n' times, outer loop iterates 'm' times and total loop iterates (m×n) times.
Given program,
for j = 1 to m
k = 1 to n
statement in the body of innerloop. None contain branching statements that lead outside the loop.
next j
next k
Here, in the inner loop k iterates from 1 to n that is n times and j iterates from 1 to m that is m times.
Finally, the whole loop iterates m×n times because for every value of j from 1 to m, k value increases from 1 to n.
So, the inner loop iterates 'n' times, outer loop iterates 'm' times and total loop iterates (m×n) times.
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For the graph y=4 find the slope of a line that is perpendicular to it
04
00
Oundefined
01/14
NEXT QUESTION
ASK FOR HELP
The slope of a line that is perpendicular to it Undefined.
What is slope?
A line's steepness can be determined by looking at its slope. Slope is calculated mathematically as "rise over run" (change in y divided by change in x).
That is the only slope which cannot be defined by a number.
A horizontal line with a slope of 0 has a change in y that is always 0 for any change in x.
As long as x is not 0, m=03, 08, 0x.
The line that runs perpendicular to this is vertical and has a "undefined" slope. We cannot divide by zero since the change in x for every change in y is always 0.
m=60,−50,y0 etc.
The slope remains undefined.
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A water taxi carries passengers from harbor to another. Assume that weights of passengers are normally distributed with a mean of 198 lb and a standard deviation of 42 lb. The water taxi has a stated capacity of 25 passengers, and the water taxi was rated for a load limit of 3750lb. Complete parts​ (a) through​ (d) below.
a=Given that the water taxi was rated for a load limit of 3750 lb, what is the maximum mean weight of the passengers if the water taxi is filled to the stated capacity of 25 ​passengers? the maximum mean weight is?
b=If the water taxi is filled with 25 randomly selected​ passengers, what is the probability that their mean weight exceeds the value from part​ (a)?he probability is?
c=If the weight assumptions were revised so that the new capacity became 20 passengers and the water taxi is filled with 20 randomly selected​ passengers, what is the probability that their mean weight exceeds 187.5 ​lb, which is the maximum mean weight that does not cause the total load to exceed 3750 ​lb? the probability is?
a) The maximum mean weight of the passengers is 187.5 lb.,b) The probability that the mean weight exceeds 187.5 lb is 0.0062.
The mean weight of each passenger is 198 lb and the standard deviation is 42 lb. The water taxi has a stated capacity of 25 passengers and a load limit of 3750 lb. The maximum mean weight of the passengers is 187.5 lb, which is determined by the load limit of 3750 lb divided by the stated capacity of 25 passengers. The probability that the mean weight exceeds 187.5 lb is 0.0062, which is calculated using the normal distribution table. If the capacity is revised to 20 passengers, the maximum mean weight is still 187.5 lb. The probability of the mean weight exceeding 187.5 lb is still 0.0062. The probability that an individual passenger exceeds the maximum load limit of 3750 lb is 0.0228, which is calculated using the normal distribution table.
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Escriba la ecuación en el formato y = mx + b, dada la siguiente información: Pasa por el punto A(7,2) es paralela a
3x-y=8
Answer:
El ecuación es y = 3x + 2
Step-by-step explanation:
Lo primero que tienes que hacer es poner 3x - y = 8 en la forma y = mx + b para para saber que 'm'. Porque la ecuación que intentamos escribir es paralela a 3x - y = 8, entonces tienen la misma pendiente (m).
3x - y = 8
-3x -3x
-y = -3x + 8
/-1 /-1 /-1 dividir todo por -1 para hacer 'y' positivo.
y = 3x - 8
Y como sabemos que la ecuación pasa por (7, 2), sabemos que pasa por 2 en el y-axis (vertical). Entonces 2 es el intercepto en y (b).
Problem 2 (D.16 Exercise 5, p 511). Forty rats are placed at random in a house having 4 rooms. There is one door between rooms 1 and 2, one door between rooms 1 and 4, one door between rooms 2 and 4, one door between rooms 2 and 3, and two doors between rooms 3 and 4 see the book for a picture). There are no doors between rooms 1 and 3. After each minute, a rat may change rooms, or stay still. All possibilities are equally likely. So, here is the transition matrix: [1/3 1/4 0 1/5 ] 1/3 1/4 1/4 1/5 0 1/4 1/4 2/5 1/3 1/4 1/2 1/5 Predict the long-term distribution of rats. What is the long-term probability that a given marked rat is in room 4? You must use the method of eigenvalues and eigenvectors.
By using the method of eigenvalues and eigenvectors, the long-term probability that a given marked rat is in room 4 is 1/3
Here we have given that Forty rats are placed at random in a house having 4 rooms. There is one door between rooms 1 and 2, one door between rooms 1 and 4, one door between rooms 2 and 4, one door between rooms 2 and 3, and two doors between rooms 3 and 4 see the book for a picture). There are no doors between rooms 1 and 3. After each minute, a rat may change rooms, or stay still.
And we need to find the the long-term probability that a given marked rat is in room 4.
By using the eigenvalues method, we know that
Room 4 has the 3 exits that is from Room 1, Room 2 and Room 3.
So, here we have 3 possibilities for the exit so, the rat can choose any one these three ways,
So, the probability can be written as,
=> 1/3
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PLEASE HELP ME ASAP!!!
Consider the following quadratic function.
The required equation of the given function in the respective form is given as g(x) = 2(x - 4)² - 7, and the vertex is (4, -7).
What is the graph?The graph is a demonstration of curves that gives the relationship between the x and y-axis.
Here,
g(x) = 2x² - 16x + 25
g(x) = 2[x² - 8x] + 25
g(x) = 2[x² - 8x + 16 -16] + 25
g(x) = 2(x - 4)² - 32 + 25
g(x) = 2(x - 4)² - 7
Now, the vertex is given as (h, k) = (4, -7).
And the graph of the given function is shown.
Thus, the required equation of the given function in the respective form is given as g(x) = 2(x - 4)² - 7, and the vertex is (4, -7).
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Compute Shannon's %NMC. Round your answer to the nearest percent. For example, if your
computed answer is 99.4% your answer rounds down to 99%. If your computed answer is
99.5% you should round up to 100%.
The Shannon's %Net market contribution = 40%
What is NMC (Net market contribution)?
Net market contribution (NMC) is equal to total sales revenue less total costs (except market costs) minus market costs.
Cost overall (not including market expenses):
150000 packaged (in cans and bottles).
Kegs =80000
WIP Loss/Reduction = 42000
Labor Contracts = 9000
Total Direct Labor: 240000
18000 inbound pounds
180000 in compensation
Total= 719000
Consumer advertising costs 26850 on the market.
Marketing for trade = 31761
Promotional Sales = 18000
Total=76611 Net market contribution=1320271-719000-76611=$ 524660 %NMC=524660/1320271=0.3974=39.74%=40%
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Count the best-case number of + operations performed by the following pseudocode segment. Assume that all possible data sets are equally likely. Preconditions: X = {x1, x2, x3, x4, x5} ⊆ {10, 20, 30, 40, 50, 60, 70, 80}, where x1 < x2 < x3 < x4 < x5. t ← 0 i ← 1 while t < 101 do t ← t + xi i ← i + 1
For the best cases there will be 6+operations, The number of operations are best cases 6 and the worst cases are 10.
Given that,
The following pseudocode snippet performs the maximum number of + operations. Assume that the probability of each potential piece of data is equal. Preconditions: X = {x₁, x₂, x₃, x₄, x₅} ⊆ {10, 20, 30, 40, 50, 60, 70, 80}, where x1 < x2 < x3 < x4 < x5. t ← 0 i ← 1 while t < 101 do t ← t + xi i ← i + 1
We know that,
Here,
X = {x₁, x₂, x₃, x₄, x₅} ⊆ {10, 20, 30, 40, 50, 60, 70, 80}
By doing the iteration method
Iteration process till 4th iteration we get 6
Therefore, For the best cases there will be 6+operations, The number of operations are best cases 6 and the worst cases are 10.
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What properties of equality do you need to use to solve the equation 2.1x−1.6=6.8? Solve the equation. Question content area bottom Part 1 What properties of equality do you need to use to solve the equation? Select all that apply. A. The Division Property of Equality B. The Multiplication Property of Equality C. The Subtraction Property of Equality D. The Addition Property of Equality Part 2 The solution is enter your response here. (Type the value of x.)
it is 6.1
Step-by-step explanation:
What is 1 1/2 + 5/7?
No matter what I've tried, nothing works. Hopefully you can help me
Answer:
see explanation
Step-by-step explanation:
given Δ ABD is isosceles with AB = AD then base angles are congruent, so
∠ ADB = ∠ ABD = 72°
then
∠ BAD = 180° - (72 + 72)° = 180° - 144° = 36° ( sum of angles in a triangle )
given Δ ABC is isosceles , then
∠ BCD = ∠ BAD = 36°
the exterior angle of a triangle is equal to the sum of the 2 opposite interior angles.
∠ ADB is an exterior angle of Δ BCD , then
∠ BCD + ∠ DBC = ∠ ADB
36° + ∠ DBC = 72° ( subtract 36° from both sides )
∠ DBC = 36°
So ∠ BCD = ∠ DBC = 36°
Thus the base angles of Δ BCD are congruent , then
Δ BCD is isosceles with BD = CD
Please answer the two following questions.
The transformed graph is attached
The domain is (-∝, ∝) and the range is (-3, ∝)
How to transform the graph?From the question, we have the following parameters that can be used in our computation:
y = eˣ
To transform the graph to y = e⁻ˣ - 3, we make use of the following rules:
Reflect across the y-axisShift down by 3 unitsSee attachment for the graph of y = e⁻ˣ - 3
The domain and the rangeFrom the attached graph, we have:
The graph extends across the x-axis i.e. domain = (-∝, ∝)
The graph opens upward from y > -3 i.e. range = (-3, ∝)
Read more about domain and range at
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A piece of cardboard is 5 centimeters longer than it is tall. The area of the piece of cardboard is 66 square centimeters. How many centimeters long is the piece of cardboard?
Answer:
12.244 cm long
Step-by-step explanation:
The area of a rectangle is equal to its height times its width, so if the piece of cardboard has an area of 66 square centimeters and is 5 centimeters longer than it is tall, we can write the following equation to represent the dimensions of the piece of cardboard:
height * (height + 5) = 66
To solve this equation, we can first divide both sides of the equation by the height of the piece of cardboard to get:
height + 5 = 66 / height
We can then subtract 5 from both sides of the equation to get:
height = 66 / height - 5
We can then multiply both sides of the equation by the height to get:
height^2 = 66 - 5 * height
We can then use the quadratic formula to solve for the height of the piece of cardboard:
height = (-5 +/- sqrt(5^2 - 4 * 1 * -66)) / (2 * 1)
The value of the square root in the quadratic formula must be positive, so we take the positive value for the height:
height = (-5 + sqrt(5^2 + 4 * 1 * 66)) / (2 * 1)
This simplifies to:
height = (5 + sqrt(61)) / 2
The height of the piece of cardboard is approximately 7.244 centimeters. Since the piece of cardboard is 5 centimeters longer than it is tall, it must be 5 + 7.244 = 12.244 centimeters long.
Therefore, the piece of cardboard is approximately 12.244 centimeters long.