The general solution of the system of differential equations is given by the two equations:
y = ±e^(4x+C1)
x = ±e^(-y/2+C2)
where the ± signs indicate the two possible solutions depending on the initial conditions.
What is the solution of the system of differential equations. = 4y 3 = -x 2?To solve the system of differential equation, we first use the given equations to find the general solution for each variable separately.
This is done by isolating the variables on one side of the equation and integrating both sides with respect to the other variable.
Once we have the general solutions for each variable, we can combine them to form the general solution for the system of differential equations.
This is done by substituting the general solution for one variable into the other equation and solving for the other variable.
The resulting general solution contains two possible solutions, each with its own constant of integration. The choice of which solution to use depends on the initial conditions of the problem.
To solve the system of differential equations:
dy/dx = 4y
dx/dy = -x/2
Finding the general solution for the first equationThe first equation can be written as:
dy/y = 4dx
Integrating both sides:
ln|y| = 4x + C1
where C1 is the constant of integration.
Taking the exponential of both sides:
|y| = e^(4x+C1)
Simplifying by removing the absolute value:
y = ±e^(4x+C1)
where ± represents the two possible solutions depending on the initial conditions.
Finding the general solution for the second equationThe second equation can be written as:
dx/x = -dy/2
Integrating both sides:
ln|x| = -y/2 + C2
where C2 is the constant of integration.
Taking the exponential of both sides:
|x| = e^(-y/2+C2)
Simplifying by removing the absolute value:
x = ±e^(-y/2+C2)
where ± represents the two possible solutions depending on the initial conditions.
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Quick algebra 1 question for 50 points!
Only answer if you know the answer, quick shout-out to Yeony2022, tysm for the help!
Answer:
d: y=5x
Step-by-step explanation:
Since we know a few points: (2,10), (3,15), (4,20), we can use two of them to first find the slope. The equation for the slope is (Increase in y)/(increase in x).
1. Slope between (2,10) and (3,15) is 5/1 which is 5. So the slope will be 5.
2. With the slope, we know that our equation will be: y=5x+b. To find b, we can replace x and y with another point. 20 = 5(4)+b. --> 20 = 20 +b. Now we know that b equals 0.
Our final equation will be y = 5x.
RECHECKING!
I'll recheck whether my answer is right or not using Desmos (A online graphing calculator).
I added an image down below, and I drew the points in blue. This shows that since the graph crosses all three points, it is correct.
WHY y=1/5x IS NOT CORRECT (based on your comments)
If you look at the points, the relationship between x and y is not 1/5. If you multiply something from x, it should become 10.
--> 2 times __ = 10
--> 3 times __ = 15
--> 4 times __ = 20
And clearly, __ will be 5. If it becomes 1/5, we won't get the right number of y.
This is why y=1/5 is not correct.
Hope this helps!
Question 4 of 5
Which values are solutions to x < 7?
OA. x= 1 and x = 6
OB. x= 8 and x = 12
OC. x = 6 and x = 12
OD. x=8 and x = -1
HELP ASP please
. A special safe lets you choose from 9 symbols for a 3-symbol-long pass code. You may enter a symbol any number of times. How many potential pass codes are there
The question is an illustration of combination and there are 729 potential pass codes available
How to determine the number of potential pass codes?The given parameters are
Symbols available = 9
Length of pass code = 3
From the question, we understand that a symbol may be entered any number of times.
This means that each of the 9 available symbols can be used three times
So, the number of potential pass codes is
Passcodes = 9 * 9 * 9
Evaluate the product
Passcodes = 729
Hence, there are 729 potential pass codes available
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Find the solution of the differential equation that satisfies the given initial condition. du dt = 2t + sec2(t) 2u , u(0) = â2
It looks like the equation is
[tex]\dfrac{du}{dt} = 2t + \dfrac12 \sec^2(t) u[/tex]
with initial value [tex]u(0) = \frac\pi2[/tex].
Rearrange the equation to
[tex]\dfrac{du}{dt} - \dfrac12 \sec^2(t) u = 2t[/tex]
Multiply both sides by the integrating factor
[tex]\displaystyle \mu = \exp\left( \int -\frac12 \sec^2(t)\,dt\right) = \exp\left(-\frac12\tan(t)\right)[/tex]
and solve for [tex]u[/tex].
[tex]\implies e^{-\frac12 \tan(t)} \dfrac{du}{dt} - \dfrac12 \sec^2(t) e^{-\frac12 \tan(t)} u = 2t e^{-\frac12 \tan(t)}[/tex]
[tex]\implies \dfrac{d}{dt}\left[e^{-\frac12 \tan(t)} u\right] = 2t e^{-\frac12 \tan(t)}[/tex]
By the fundamental theorem of calculus, integrating both sides yields
[tex]e^{-\frac12\tan(t)} u = e^{-\frac12\tan(t)} u \bigg|_{t=0} + \displaystyle \int_{\xi=0}^{\xi=t} 2\xi e^{-\frac12 \tan(\xi)} \, d\xi[/tex]
[tex]\implies e^{-\frac12\tan(t)} u = 1\times\dfrac\pi2 + \displaystyle \int_{0}^{t} 2\xi e^{-\frac12 \tan(\xi)} \, d\xi[/tex]
[tex]\implies \boxed{\displaystyle u = \frac\pi2 e^{\frac12\tan(t)} + 2e^{\frac12\tan(t)} \int_0^t \xi e^{-\frac12 \tan(\xi)} \, d\xi}[/tex]
The design pattern on a blanket has 3 circles followed by 4 triangles. this pattern repeats across the entire blanket. move a number or word to each blank to describe the ratio relationship between the circles and triangles.
The ratio between the circle and triangle is described as 3:4.
What is ratio?An ordered pair of numbers a and b, written as a / b, is a ratio if b is not equal to 0.A proportion is an equation that sets two ratios at the same value.In this section, the terms ratio and proportion are defined. Both ideas play a significant role in mathematics. Numerous examples where the concept of the ratio is highlighted may be found in daily life, such as the rate of speed (distance/time) or price (rupees/meter) of a substance.An equation for proportion establishes the equality of the two provided ratios.Only when the two numbers in a ratio have the same unit can they be compared. Ratios are used to compare two objects. A ratio is indicated by the symbol A fraction, like 2/5, can be used to represent a ratio. In our daily lives, we come across many comparisons, or ratios.As a result, the ratio can be expressed in three distinct ways, including:a to ba : ba/bKnow more about ratio and proportion numerical
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Which equation represents the line that is perpendicular to y = 1/4 and passes through (-6,-9)? OA X=-9 OB. Oc y=-9 D. y=-4
Find a polynomial function of degree 7 with a leading coefficient of 1 and with - 3 as a zero of multiplicity 3,0 as a zero of multiplicity 3, and 3 as a zero of multiplicity 1.
The least polynomial in standard form is defined by the function f(x) = x⁷ + 6 · x⁶ - 54 · x⁴ - 81 · x³.
How to determine the least polynomial given a set of roots and a leading coefficient
Polynomials can be expressed as a product of binomials of the form (x - r) multiplied by a leading coefficient. The least polynomial contain the number of roots presented in statement, whose factor form is shown below:
f(x) = 1 · (x + 3)³ · x³ · (x - 3)
f(x) = (x + 3)³ · (x⁴ - 3 · x³)
f(x) = (x³ + 9 · x² + 27 · x + 27) · (x⁴ - 3 · x³)
f(x) = x⁷ + 9 · x⁶ + 27 · x⁵ + 27 · x⁴ - 3 · x⁶ - 27 · x⁵ - 81 · x⁴ - 81 · x³
f(x) = x⁷ + 6 · x⁶ - 54 · x⁴ - 81 · x³
The least polynomial in standard form is defined by the function f(x) = x⁷ + 6 · x⁶ - 54 · x⁴ - 81 · x³.
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15. The sides of a parallelogram are 100 m each and the length of the longest diagonal is 160 m. Find the area of the parallelogram.
Answer:
Area of parallelogram = 9600 m²
Step-by-step explanation:
• We can find the measure of angle x using the cos rule:
[tex]160^2 = 100^2 +100^2 -2(100)(100) \cdot cos (x)[/tex]
Make x the subject of the equation:
⇒ [tex]20000 \cdot cos(x) = 100^2 +100^2 - 160^2[/tex]
⇒ [tex]cos(x) = -0.28[/tex]
⇒ [tex]x = cos^{-1} (-0.28)[/tex]
⇒ [tex]x = 106.26 \textdegree[/tex]
• Now we can find the area of one the triangles formed using the formula:
[tex]Area =\frac{1}{2} ab \cdot sin \theta[/tex]
where a and b are two sides of a triangle, and θ is the angle between them (angle x).
Substituting the values:
Area of one triangle = [tex]\frac{1}{2}[/tex] × (100)(100) × sin(106.26°)
= 4800 m²
• Since the parallelogram is formed by two such triangles, we have to double the area of the triangle to find the parallelogram's area:
Area of parallelogram = 2 × 4800
= 9600 m²
Which graph shows y < x² +1?
O
4) Intro
•
What is inequality?
In Algebra, inequality shows the relation between two expressions using the inequality symbol.
The graph shows the equation y<x^2+1 is attached
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The graph at option 1 shows the given inequality y < x² + 1. The domain and range of the given inequality is {x: x ∈ (-∞, ∞)} and {y: y ∈ [1, ∞)}.
How to graph an inequality?The steps to graph an inequality equation are:
Solve for the variable y in the given equationGraph the boundary line for the inequalityShade the region that satisfies the inequality.Calculation:The given inequality is y < x² + 1
Finding points to graph the boundary line by taking y = x² + 1:
When x = -2,
y = (-2)² + 1 = 4 + 1 = 5
⇒ (-2, 5)
When x = -1,
y = (-1)² + 1 = 2
⇒ (-1, 2)
When x = 0,
y = (0)² + 1 = 1
⇒ (0, 1)
When x = 1,
y = (1)² + 1 = 2
⇒ (1, 2)
When x = 2,
y = (2)² + 1 = 5
⇒ (2, 5)
Plotting these points in the graph forms an upward-facing parabola.
So, all the points above the vertex of the parabola satisfy the given inequality. Thus, that part is shaded.
From this, the graph at option 1 is the required graph for the inequality y < x² + 1. The boundary line is dashed since the inequality symbol is " < ".
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20 POINTS
please help and have a good weekend
A = 6 cm; B = 3 cm; C = 8 cm; D = 10 cm
The Surface area of the prism = 120 cm².
How to Find the Surface Area of Triangular Prism?Surface area = Area of A + B + C + D
The side lengths are:
A = 6 cm
B = 3 cm
C = 8 cm
D = 10 cm
Surface area of the prism = 2(area of triangular face) + area of rectangle 1 + area of rectangle 2 + area of rectangle 3
Area of triangular face = 1/2(b)(h) = 1/2(8)(6) = 24 cm
Area of rectangle 1 = (length)(width) = (10)(3) = 30 cm²
Area of rectangle 2 = (length)(width) = (6)(3) = 18 cm²
Area of rectangle 3 = (length)(width) = (8)(3) = 24 cm²
Surface area of the prism = 2(24) + 30 + 18 + 24 = 120 cm².
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3 + 2 x 7 - 30(^2)
help me
A couple quick algebra 1 questions for 50 points!
Only answer if you know the answer, quick shout-out to Subtomex0, tysm for the help!
Answer:
Question 6) Neither (Not an inverse variation)
Question 7) Inverse variation
Question 8) Inverse variation
Step-by-step explanation:
These are the two formulas we need to know prior to solving the question.
Direct variation: c times x
Inverse variation: c times 1/x --> c/x
- Here, c is the constant
6. y = x + 7
This is neither. It is not in the form of a direct, or inverse variation, so it does not belong in any category.
7. 14 = xy
This equation is an inverse variation.
xy = 14
y = 14/x
14 is the constant, and the format is the same as for an inverse variation.
8. x = -8/y
This equation is an inverse variation.
yx = -8
y = -8/x
-8 is the constant, and the format is the same as for an inverse variation.
So we can conclude that questions 7 and 8 is an inverse variations, while question 6 isn't.
General forms
Direct
y=KxInverse
y=k/xso
#6
y=x+7Not inverse and direct also as 7 is present
#7
14=XYy=14/xYes
#8
x=-8/yy=-8/xYes
Another copy machine also has the ability to reduce image dimensions, but by a different percentage. This graph shows
the results found when copying a design x times. Use the graph to write the equation modeling this relationship.
-10
2 3 4
6 7 8
Enter the correct answer in the box by feplacing the values of a and b.
The correct values for a and b which models this relationship through an exponential function are 8 and 2 respectively.
What is an exponential function?An exponential function can be defined as a mathematical function whose values are generated by a constant that is raised to the power of the argument. Mathematically, an exponential function is represented by this formula:
f(x) = eˣ
How to write an equation modeling this relationship?Based on the information provided, we can logically deduce that the other copy machine can reduce image dimensions by a different percentage and this is defined by the following exponential function:
f(x) = abˣ
From the graph (see attachment), we have;
When x = 0, y = f(x) = 8. Thus, f(0) = 8
f(x) = abˣ
8 = ab⁰
8 = a × 1
a = 8.
Also, from the graph (see attachment), we have;
When x = 1, y = f(x) = 4
f(x) = abˣ
4 = 8b¹
4 = 8 × b
4 = 8b.
b = 8/4
b = 2.
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If $25, 000 is invested into an account paying 4.5% per year, compounded quarterly, how much money can be withdrawn from the account every two months for the next 6 years?
The money can be withdrawn from the account every two months for the next 6 years is $20, 625, 000
What is compound interest?
The formula for compound interest is given as;
[tex]A = P ( 1 + \frac{r}{n} )^n^t[/tex]
P = principal interest = $25, 000
r = rate = 4. 5%
n = 2 months
t = 6 years
A = [tex]25, 000 ( 1 + \frac{4. 5}{2} )^2 ^*^6[/tex]
A = [tex]25, 000(1 + 0. 75)^1^2[/tex]
A = [tex]25, 000( 825. 00)[/tex]
A = $20, 625, 000
Thus, the money can be withdrawn from the account every two months for the next 6 years is $20, 625, 000
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Ms. johnson owns a spice shop. she combines 5 pounds of dried rose hips and 1 pound of five-spice powder to make a new seasoning. how much should she charge for 1 ounce of the new seasoning if she can sell the dried rose hips for $8 per pound and the five-spice powder for $11 per ounce?
Given the 6 pound weight of the new seasoning having ingredients that cost $8, and $11, the amount she should charge for 1 ounce is $0.53125
Which method can be used to calculate the price of the new seasoning?The ingredients in the new seasoning are;
5 pounds of dried rose hips + 1 pounds of five-spice powder
Cost of the ingredients are;
1 pound of dried rose hips = $8
1 pound of five-spice powder = $11
The cost of combined spice, C, is therefore;
C = $8 × 5 + $11 × 1 = $51
The weight of the new combined seasoning = (5 + 1) pounds = 6 pounds
Cost of 1 pound of the new seasoning is therefore;
[tex]1 \: lb = \frac{51}{6} = 8.5[/tex]
1 lb of the new seasoning = $8.51 lb = 16 ouncesTherefore;
[tex]1 \: ounce = \frac{8.5}{16} = \mathbf{0.53125}[/tex]
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Answer: $2.25
Step-by-step explanation:
5 lbs = 80 oz, so the mixture is worth
80(8/16) + 16(11) = 96x
x = $2.25/oz
Hope this helps!!
the area of a rectangle is (16x^2 - 9y^2) square units
HELP ME
ANY GENIUS OUT THERE>
Answer:
angle x equals 25
Step-by-step explanation:
130 + 25 = 155
180 - 155 = 25
Triangles always add up to 180°
Answer:
it is so easy the answer is 25
Step-by-step explanation:
The total sum of the traingle is 180 degree. So frist add both the numbers in the traingle and then subtract it by 180 degree
PLESSEE HELP ME ASAPP
Answer:
option A. is the midpoint of the graph
Given a population comprised of 30 bats, 15 gloves, and 60 balls, if sampling is random and one at a time without replacement, what is the probability of obtaining a bat and a ball in that order if two objects are sampled from the populationGroup of answer choices
The probability would be 3/125 in the given conditions.
As there are 105 items, of which 30 are bats. The probability for that draw is 30/105.
There are now 104 items remaining, of which 15 are gloves. The probability of getting a glove now is 15/104.
There are now 103 items remaining, of which 60 are balls. The probability of getting a ball now is 60/103.
The probability of all three occurring = product of three
15/104* 30/105* 60/103
= 3/125
Hence the probability is 3/125.
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Jim is driving to Denver. Suppose that the distance to his destination (in miles) is a linear function of his total driving time (in minutes). Jim has 61 miles to his destination after 41 minutes of driving, and he has 42.3 miles to his destination after 63 minutes of driving. How many miles will he have to his destination after 81 minutes of driving
Jim has to travel 26.82 miles more to reach Denver.
What is the equation of a line?
Let the first point be = (x₁ , y₁)
Let the second point be = (x₂ , y₂)
m(slope) = (y₂ - y₁) / (x₂ - x₁)
Equation of line = (y - y₁) = m(x - x₁)
Let the no. of miles be plotted on X-axis
Let the driving time be plotted on Y-axis
First point = (61 , 41)
Second point = (42.3 , 63)
Slope of the linear function = (63 - 41)/(42.3 - 61) = -1.17
Equation of the line = (y - 41) = -1.17(x - 61)
Given time of driving = 81 min
Number of miles = (81 - 41) = -1.17(x - 61)
40/-1.17 = x - 61
-34.18 + 61 = x
x = 26.82
Hence, Jim has to travel 26.82 miles more to reach Denver.
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Find the area of a circle with diameter, d = 3.3m. give your answer rounded to 1 dp.
Answer:
Step-by-step explanation:
the area of a circle is πr²
r being the radius, which is equal to half the diameter
3.3/2= 1.65
π(1.65)²=8.6m
The acts in a talent competition consist of 4 instrumentalists, 10 singers, and 6 dancers. If the acts are ordered randomly, what is the probability that a dancer performs first
Answer:
[tex] \frac{6}{20} = \frac{3}{10?} [/tex]
Step-by-step explanation:
total amount of acts = 4+10+6 = 20
amount of dancers = 6
probability of picking a dancer = number of dancers / total number of talents = 6/20 = 3/10
The probability that a dancer performs first is 3/10.
What is probability?Probability refers to potential. A random event's occurrence is the subject of this area of mathematics.
The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events.
The degree to which something is likely to happen is basically what probability means.
Given:
The acts in a talent competition consist of 4 instrumentalists, 10 singers, and 6 dancers.
Total participants = 4+ 10 +6 = 20
So, the probability that a dancer performs first
= 6 /20
= 3/10
Hence, the probability that a dancer performs first is 3/10.
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Select the correct answer. A line t runs upward to the right through two parallel horizontal lines r and s, forming 8 angles numbered from 1 to 8. Given: Transversal t passes through parallel lines r and s. Prove: ∠3 ≅ ∠6 ∠4 ≅ ∠5 Statement Reason 1. r || s given
2. ∠3 ≅ ∠7 ∠4 ≅ ∠8 For parallel lines cut by a transversal, corresponding angles are congruent.
3. What is the next step in the proof? Choose the most logical approach.
A. Statement: ∠1 ≅ ∠8 and ∠2 ≅ ∠7 Reason: Congruent Supplements Theorem
B. Statement: m∠3 + m∠4 = 180° and m∠7 + m∠8 = 180° Reason: Linear Pair Theorem
C. Statement: m∠3 + m∠5 = 180° and m∠4 + m∠6 = 180° Reason: definition of supplementary angles
D. Statement: ∠7 ≅ ∠6 and ∠8 ≅ ∠5 Reason: Vertical Angles Theorem Reset Next
The the next step in the proof is D. Statement: ∠7 ≅ ∠6 and ∠8 ≅ ∠5 Reason: Vertical Angles Theorem.
How to illustrate the angle?It should be noted that vertical angles are the angles that are opposite each other when the two line cross.
In this case, 7 ≅ ∠6 and ∠8 ≅ ∠5 Reason: Vertical Angles Theorem.
In conclusion, the correct option is D.
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A school offers three subjects: math, art and science. at least 80% of students study both math and art. at least 80% of students study both math and science. prove that at least 80% of students who study both art and science, also study math.
The proof to show that at least 80% of students who study both art and science, also study math is illustrated below.
How to illustrate the proof?From the information given, it was stated that at least 80% of students study both math and art and that at least 80% of students study both math and science.
Therefore, about (100 - 80) 20% doesn't study the combination of both subjects.
Here, those students who study both art and science, and also study math will be:
= 100% - 20%
= 80%
Therefore, at least 80% study the subjects.
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can you help me with this
Two students compared the scores on their math tests. Their results on 9 tests are shown in the box plots below.
Compare the box plots. Which statements are true? Check all that apply.
Ben has the smaller range of test scores.
The lower quartile and upper quartile of Nadia’s data are both higher than the lower quartile and upper quartile of Ben’s data.
The median of Nadia’s data is equal to the median of Ben’s data.
Nadia had the highest score on a test.
Ben’s lowest score was equal to Nadia’s lowest score
The statements that are true for the box plots are:
C. The median of Nadia’s data = median of Ben’s data.
D. Nadia had the highest score on a test
How to Find the Median of a Box Plot?The median of a data set is the center value, which is indicated on a box plot by a vertical line that divides the box.
Given the two box plots, we can conclude that:
Median for the data of Nadia is 92, which is the same for Ben.
The extreme whisker to the right for Nadia is at 100, which is the highest value for Nadia while that of Ben is 99.
The true statements are:
C. The median of Nadia’s data = median of Ben’s data.
D. Nadia had the highest score on a test
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Answer:
"C" and "D"
Step-by-step explanation:
took test on edge 2023
A management consulting firm recommends that the ratio of middle-management salaries to management trainee salaries be 8:7. Using this recommendation, what is the annual middle-management salary if the annual management trainee salary is $17,000? (Round your answer to the nearest dollar.)
The management salary is $19,429
What is ratio?
Ratio is the quantitative relationship between two numbers which shows the number of times an amount or a number is contained within the other amount or number.
In this case, a ratio of 8:7 shows the number of times the management salaries is contained within the management trainee salary
The management salary has a value of 8 whereas the management trainee salary has a value of 7 as shown below:
salary ratio=management salary/management trainee salary
salary ratio=8/7
management salary=unknown(assume it is X)
management trainee salary=$17,000
8/7=X/$17,000
cross multiply
8*$17,000=7*X
X=8*$17,000/7
X=$19,429
The fact that annual management trainee salary is $17,000 means that the management salary is $19,429 annually.
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helpppppppp. look at pic
Answer:
h (1) = $140
Step-by-step explanation:
10x + 150
-10 -10
150-10
=140
so the answer must be $140
So it is definitely the first one
H(12)=$270
A pizza company is building a rectangular solid box to be able to deliver personal pan pizzas. The pizza company wants the volume of the delivery box to be 756 cubic inches. The length of the delivery box is 6 inches less than twice the width, and the height is 2 inches less than the width. Determine the width of the delivery box.
The width of the delivery box is 9 inches.
Given that volume of delivery box is 756 cubic inches,length is 6 inches less than twice the width, height is 2 inches less than width.
let width be x
Length = 2x- 6
Height = x - 2
Width = x
We can find the value of x by putting values one by one and then we will be able to get the value of x which is width.
Because the delivery box is cuboid in shape so the volume is equal to product of length,breadth and height.
When x=7: Volume=7 x (7 - 2) x (2x7 - 6) = 7 x 5 x 8 = 280
When x=9,Volume will be 9 x (9 - 2) x (2x9 - 6) = 9 x 7 x 12 = 756
Therefore, the width of the delivery box is 9 inches.
Hence the width of the delivery box is 9 inches.
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WILL MAKE BRAINLIEST!! If m
Answer:
98
Step-by-step explanation:
m<KJU + m<UJI = m<KJI
this equation will help us find the value of x therefore the value of m<KJU
Rewrite the equation using the expressions
104 + x + x + 76 = 168 add like terms
180 + 2x = 168 subtract 180 from both sides
2x = -12 divide both sides by 2
x = -6 to find the value of m<KJU replace x with -6
104 + (-6) = 98