To find a nonzero solution to a system of homogeneous equations, the corresponding matrix must be singular, meaning it has no inverse and its determinant is zero, allowing for the existence of eigenvectors corresponding to the eigenvalue λ = 0.
When we have a system of homogeneous equations, it means that all the equations in the system have a right-hand side of zero. Such a system can be written as Ax = 0, where A is the coefficient matrix and x is the column vector of variables.
To find a nonzero solution, we need to find a value of x such that Ax = 0, but x ≠ 0.
If A is a singular matrix, it means that it has no inverse, and its determinant is zero. This implies that the rows of A are linearly dependent, which in turn means that the columns of A are also linearly dependent. In other words, some of the columns of A can be expressed as linear combinations of the others.
When we multiply a singular matrix A by any nonzero vector x, we get a linear combination of its columns that equals zero, since the columns of A are linearly dependent.
Therefore, we can always find a nonzero vector x that satisfies Ax = 0. This nonzero vector x is called an eigenvector corresponding to the eigenvalue λ = 0, which is the only eigenvalue of a singular matrix.
Hence, finding a nonzero solution to a system of homogeneous equations requires that the matrix corresponding to this system is singular because only singular matrices have eigenvectors corresponding to the eigenvalue λ = 0, which are the solutions to the homogeneous equation Ax = 0.
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A plane ticket to Barcelona cost £175 the price decreases by 6% work out the new price of the plane ticket
Answer:
£164.50
Step-by-step explanation:
If the discount is 6%, then the discounted price is 94% of the original price.
95% of £175 = 0.94 × £175 = £164.50
Which of the following explains why it is easier to reject the null hypothesis with a one-tailed test than with a two-tailed test with all the same parameters?
a. Because the standard deviation in a one-tailed test is larger than that for a two-tailed test
b. Because z-scores are calculated differently in a one-tailed test
c. Because the critical region is all on one side in a one-tailed test and needs to be split between the two tails in a two-tailed test
d. Because a two-tailed test uses a bimodal distributio
Because the critical region is all on one side in a one-tailed test and needs to be split between the two tails in a two-tailed test.
What is the difference between a one-tailed test and a two-tailed test in hypothesis testing?In a one-tailed test, the alternative hypothesis is directional and the critical region is all on one side of the distribution. In a two-tailed test, the alternative hypothesis is non-directional and the critical region is split between the two tails of the distribution.
Why is it easier to reject the null hypothesis with a one-tailed test than with a two-tailed test?It is easier to reject the null hypothesis with a one-tailed test because the critical region is all on one side of the distribution, making it more likely to observe a significant difference and reject the null hypothesis. In contrast, the critical region is split between the two tails in a two-tailed test, making it less likely to observe a significant difference and reject the null hypothesis.
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$15,600 at 3% for 2 years
Answer:10,400
Step-by-step explanation:
for each increase of one unit on the x-axis (the horizontal axis), the amount on the y-axis (the vertical axis) increases by:
For each increase of one unit on the x-axis (the horizontal axis), the amount on the y-axis (the vertical axis) increases by the slope of the line.
Slope is the amount of change in the y-axis that occurs as a result of a one-unit change in the x-axis. It is also known as the rise over run.
A positive slope indicates that the line rises from left to right, while a negative slope indicates that the line falls from left to right. If the slope is zero, the line is horizontal.
If the line slopes up from left to right, it has a positive slope. As the value of x increases by one, the value of y also increases by the slope. If the line slopes downward from left to right, it has a negative slope. As the value of x increases by one, the value of y decreases by the slope.The slope of a horizontal line is zero, and the slope of a vertical line is undefined because the x or y coordinate does not change as the other changes.
Therefore, the slope of a line represents the amount by which the value on the y-axis increases for every increase of one unit on the x-axis
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I need help
Write an equation for the table below.
Answer:
5x1x5=712343x12x67=800
Step-by-step explanation:
Answer:
y = 5x
Step-by-step explanation:
y = water used (gal)
t = time (min)
y = 5x
This means for every minute, 5 gallons of water is used
What are all of the constants in the equation n + 6 = 30?
Answer:
Simple and best practice solution for n/6=30= equation. Check how easy it is, ... n-30*6=0. We add all the numbers together, and all the variables n-180=0
Given that New Mexico has a population of about 12 people per square miles and
an area of about 120,000 square miles, what is the population of New Mexico
Mark only one oval.
10,000
1,000,000
1,440,000
2,400,000
Answer:
1440000
Step-by-step explanation:
Since for every mile, there are 12 people, we have 120000 miles. If we multiply 12 by 120000, we get 1440000.
with reference to one 20th-century war, compare and contrast the political repercussions for two countries.
Germany and Japan have been profound and long-lasting, these countries have developed in the years since the war.
How impacts of 20th-century war on two countries?The 20th century saw numerous wars that had a significant impact on the global political landscape. The two countries in question, let us assume, are Germany and Japan.
During World War II, these two countries were on the same side, the Axis powers, and were both defeated by the Allies.
As a result of this war, both countries experienced a range of political repercussions.
In the aftermath of the war,
both countries were occupied by Allied powers, with the United States occupying Japan and Germany being divided into four separate zones of occupation, each controlled by a different Allied power.
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What is the effect on the graph of the function f(x) = 2^x when f(x) is replaced with f(x − 3)? A.)translate horizontally 3 units right B.) translate vertically 3 units down C.) translate horizontally 3 units left D.) translate vertically 3 units up
Answer:
a
Step-by-step explanation:
I just did the test
Answer:
The effect on the graph of the function f(x) = 2^x when f(x) is replaced with f(x − 3) is that it translates horizontally 3 units right.
When a function is replaced with f(x - c), where c is a positive constant, the graph of the function shifts horizontally c units to the right. Similarly, if c is negative, the graph shifts horizontally c units to the left.
Therefore, replacing f(x) with f(x - 3) in the function f(x) = 2^x shifts the graph of the function horizontally by 3 units to the right.
Please mark this answer as Brainliest!!
Work out (1 + 9)² +5
Answer:
105
Step-by-step explanation:
10² is 100
100+5=105
Written as a simplified polynomial in standard form, what is the result when
(x+4)² is subtracted from 8?
The simplified form of the polynomial expression in standard form is: - x² - 8x - 8
How to express the polynomial in standard form?The standard form of a polynomial is usually expressed by writing the highest degree of terms first then the next degree and so on. The standard form of polynomial is given by, f(x) = aⁿxⁿ + aⁿ⁻¹xⁿ⁻¹ + aⁿ⁻²xⁿ⁻² + ... + a₁x + a₀, where x is the variable and a_i are coefficients.
The polynomial expression is given as (x+4)² is subtracted from 8. Thus, we have:
8 - (x+4)²
Expanding gives:
8 - (x² + 8x + 16)
= 8 - x² - 8x - 16
= - x² - 8x - 8
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Please help with these 2 problems asap !! 15 points
Answer:
11. 17
12. -16
Step-by-step explanation:
a gallon of milk is approximately 4 liters. how many gallon containers of air will it take to full one cubic meter (1 m3)? select the appropriate set-up for this calculation.
A gallon of milk is approximately 4 liters. 250 gallon containers of air will it take to full one cubic meter.
The setup for calculating how many gallon containers of air it will take to fill one cubic meter is:
1 m³ × 1000 L/m³ ÷ 4 L
A gallon of milk is approximately equal to 4 liters.
1 cubic meter (1 m³) of air is equal to 1000 liters.
One cubic foot of water contains 7.48 gallons. Dividing the total number of gallons by 7.48 will get the cubic feet equivalent.
To convert liters to gallons,
divide by 4 (since 1 gallon is approximately equal to 4 liters).
The calculation for how many gallon containers of air it will take to fill one cubic meter is:
1 m³ × 1000 L/m³ ÷ 4 L = 250 gallons.
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The sum of the deviation of the individual data elements from their mean is always_________.
a. equal to xero
b. equal to one
c. negative
d. positive
The sum of the deviations of the individual data elements from their mean is always equal to zero. Correct option is A.
This is because the deviation is defined as the difference between each data point and the mean of all the data points.
Since the mean is calculated as the sum of all the data points divided by the total number of data points, the positive deviations from the mean must be balanced out by the negative deviations from the mean. In other words, for every data point that is above the mean, there is a corresponding data point that is below the mean.
When we add up all the deviations from the mean, the positive deviations will cancel out the negative deviations, resulting in a sum of zero. This property of the sum of deviations from the mean is fundamental to many statistical concepts, such as variance and standard deviation.
Therefore, the correct answer to the question is (a) equal to zero, since the sum of the deviations from the mean is always zero.
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HELP PLEASE ¨"half-life problem¨¨ A certain computer loses half of its value every four years. If the value of the computer after 6 years is $835, what was the initial value of the computer?
Thus, the computer was initially worth $3340 as the PC will be worth $835 after six years.
what is unitary method ?In mathematics, the unitary approach entails determining the value of a single unit in a certain circumstance and using that value to resolve other issues that are connected to it. It is a quick and efficient method for handling issues involving direct proportion, indirect proportion, and other circumstances of a same nature. In disciplines including mathematics, science, business, and engineering, the unitary technique is widely employed.
given
Let V be the computer's starting value.
The computer's value drops to V/2 after four years.
The computer's value is decreased to V/2 1/2 = V/4 after six years.
We are informed that the PC will be worth $835 after six years. When we set this equal to V/4, we obtain:
V/4 = 835
Adding 4 to both sides results in:
V = 4 × 835 = $3340
Thus, the computer was initially worth $3340 as the PC will be worth $835 after six years.
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The forecast maximum temperature, in degrees Celsius, and the observed maximum temperature are recorded to determine the accuracy in the temperature prediction models used by the weather bureau.
Around noon, when the forecast temperature is about 4 degrees Celsius higher than the measured temperature, the difference is at its greatest.
what is correlation coefficient ?The strength and direction of the linear connection between two variables are measured by the correlation coefficient. Its value goes from -1 to 1, and it is represented by the symbol "r". A perfect negative correlation, or r = -1, implies that as one variable rises, the other variable falls in an exact predictable manner. There is no linear relationship between the two variables if r = 0, which implies there is no correlation between the two variables. When r = 1, there is a perfect positive correlation, meaning that when one measure rises, the other rises in an exact predictable manner.
given
The maximum temperature for a specific day is shown in the provided image alongside the predicted maximum temperature.
The blue line depicts the expected maximum temperature, while the red line depicts the actual maximum temperature.
We can see that throughout the day, the forecast temperature is typically greater than the actual temperature.
Around noon, when the forecast temperature is about 4 degrees Celsius higher than the measured temperature, the difference is at its greatest.
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Sam, Larry, and Howard have contracted to paint a large room in a house. After calculating all the material costs, which are to be paid by the homeowner, they decide that $270 would be a fair price for the 16 hours it will take to prepare, paint, and clean up. Each of the men decides that $15.00 an hour is a fair wage for the job. If three-quarters of the work will be done by Larry, how much will Larry be paid for his work on the job?
Answer:
180 dollars
Step-by-step explanation:
270 for 16 hours = 16.875 dollars for each hour
15.00 an hour will be paid to larry
larry is doing 3/4 of those 16 hours
12 hours
15 x 12
larry gets 180 dollars for 12 hours of work
hope this helps x
On a given planet, the weight of an object varies directly with the mass of the object. Suppose that an object whose mass is 5 kg weighs 25 N. Calculate the mass of another object that weighs 15 N
The weight of an object varies directly with the mass of the object therefore, the mass of the other object that weighs 15 N is 3 kg.
We can use the formula for direct variation, which states that if y varies directly with x, then y = kx, where k is the constant of variation. In this case, the weight y varies directly with the mass x, so we have:
y = kx
We are given that when x = 5 kg, y = 25 N. Substituting these values into the formula, we get:
25 = k(5)
Solving for k, we have:
k = 5
Now we can use this value of k to find the mass of another object that weighs 15 N. Let's call the mass of this object x2. We know that y2, the weight of the object, is 15 N. So we have:
15 = 5x2
Solving for x2, we have:
x2 = 3 kg
Therefore, the mass of the other object that weighs 15 N is 3 kg.
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Write the numeral for nine hundred thousand and twelve.
Answer: 900,012
Step-by-step explanation:
Answer: 900,012
this is because, nine hundred thousand is shown as 900,000 in number form
and then you just add 12 to that number, making it 900,012
If a item is $30 and the is 20% off the regular price how much would it be
Answer:
20 percent off depends on the original cost: Take the original number and divide it by 10. Double your new number. Subtract your doubled number from the original number. You have taken 20 percent off! For $30, you should have $24.
Step-by-step explanation:
Shiloh made a scale drawing of a triangular piece of art. Shiloh used a scale factor of 4 when making her drawing. If the base and the height of Shiloh's drawing are 7 inches and 2 inches respectively, what is the area of the piece of art?
The area of the piece of art is 448 square inches.
Shiloh made a scale drawing of a triangular piece of art.
Shiloh used a scale factor of 4 when making her drawing.
If the base and the height of Shiloh's drawing are 7 inches and 2 inches respectively,
what is the area of the piece of art?
Shiloh used a scale factor of 4 when making her drawing.
Therefore, the actual dimensions of the original triangular piece are four times the dimensions of the scaled drawing. Thus, the actual dimensions of the base and the height of the triangular piece are, respectively:
4 × 7 = 28 inches and 4 × 2 = 8 inches.
Using the formula for the area of a triangle,
we can find the area of the triangular piece.
The formula is given as A=12bh where b is the base and h is the height.
A = 12 × 28 × 8A = 112 × 4A = 448
Therefore, the area of the piece of art is 448 square inches.
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PLEASE SHOW FULL SOLUTIONS FOR BOTH QUESTIONS AND ONLY ANSWER IF YOU KNOW! NO CALCULUS PLEASE! THAT WOULD BE VERY APPRECIATED!!
Answer:
5. 3 cm × 2 cm × 7 cm
6. f(x) = x³ -3x² -x +3
Step-by-step explanation:
You want the dimensions of a cuboid with edges marked (x-1), (x-2), and (x+3) and a volume of 42 cm³. You also want the coefficients of a monic cubic f(x) with values f(2) = -3, f(-3) = -48, and f(-1) = f(1).
5. CuboidThe volume of the cuboid is the product of its dimensions, so we have ...
V = LWH
42 = (x -1)(x -2)(x +3)
The value of x can be found as the solution to this equation. A graphical solution is attached. It shows x=4, so the dimensions are ...
4 -1 = 3
4 -2 = 2
4 +3 = 7
The dimensions of the cuboid are 3 cm by 2 cm by 7 cm.
Note that the prime factors of 42 are 2, 3, 7, which have the required differences. You don't really need a polynomial to guess these are the dimensions.
The expanded polynomial is x³ -7x -36 = 0, so potential rational roots will be from the set {1, 2, 3, 4, 6, 9, 12, 18, 36}. An estimate of the upper bound of the real root puts it at ∛36 +√7 ≈ 5.9. We require x > 2, so the viable choices for testing are 3 and 4. x=4 is the solution.
6. CoefficientsThe remainder theorem tells you that the remainder from dividing f(x) by (x-a) is f(a). To obtain linear equations in b, c, d, we can rearrange the function to ...
bx² +cx +d = -x³ +f(x)
For x = ±1, we know the remainders f(x) are the same, so we can look at the difference of the equations for these x-values.
(b(-1)² +c(-1) +d) - (b(1)² +c(1) +d) = (-(-1)³ +f(-1)) - (-(1)³ +f(1))
b -b -c -c +d -d = 1 -(-1)
-2c = 2
We can fill in values x=2, x=-3 to get two more equations in b, c, d. The coefficients of the three equations we have for the three unknowns are shown in the second attachment. The third attachment shows the solution of these equations is (b, c, d) = (-3, -1, 3).
The table in the first attachment confirms the remainders using these coefficients.
__
Additional comment
For problem 6, we started out using synthetic division, then realized the resulting equations are the same as those developed using the rearranged form shown above. We like to let calculators and spreadsheets do the tedious arithmetic where possible.
Answer:
5)The dimensions of the rectangular prism are 3 , 7 and 2.
6) f(x) = x³ - 3x² - x + 3
Step-by-step explanation:
5) The volume of rectangular prism = l*w*h.
l*w*h= 42
(x- 1)(x -2)(x+3) = 42
We can find (x - 2) *(x + 3) using the identity (x + a)(x + b) = x² + (a +b)*x + ab
(x - 2)(x + 3) =x² + (-2 +3)x + (-3)*2
= x² + 1x - 6
(x -1)(x + 3)(x - 2) = 42
(x - 1)(x² + x - 6) = 42
x*x² + x*x - 6*x + (-1)*x² + (-1) *x + (-1)(-6) = 42
x³ + x² - 6x - x² - x + 6 - 42 = 0
x³ + x² - x² - 6x - x + 6 -42 = 0
Combine the like terms,
x³ - 7x - 36 = 0
Find the zeros of the cubic polynomial by synthetic division method.
4 1 0 -7 -36
4 16 36
1 4 9 0
x - 4 is zero of the polynomial.
Ignore the quadratic polynomial x² + 4x + 9 as it will have irrational roots(zeros) and dimensions will be always positive integer.
x - 4 = 0
x = 4
length = x - 1 = 4 - 1 = 3
Width = x + 3 = 4 + 3 = 7
Height = x - 2 = 4 - 2 = 2
The dimensions of the rectangular prism are 3 , 7 and 2.
6) f(x) = x³ + bx² + cx + d
It is given that when f(x) is dived by x + 1 and x- 1, the remainders are same.
x + 1 = 0 ; x - 1 = 0
x = -1 ; x = 1
f(-1) = f(1)
-1 + b - c + d = 1 + b + c + d
-1 -1 + b - b - c - c + d - d = 0
-2 - 2c = 0
-2c = 2
c = 2 ÷ (-2)
[tex]\boxed{c = -1}[/tex] -------------(I)
It is given that when f(x) divided by ( x - 2) it leaves a remainder (-3)
f(2) = -3
8 + 4b + 2c + d = -3
8 + 4b + 2*(-1) + d = -3 {from (I)}
8 + 4b - 2 + d = -3
4b + d = -3 + 2 - 8
4b + d = -9 -------------(II)
It is given that when f(x) divided by ( x + 3) it leaves a remainder (-48).
f(-3) = -48
-27 + 9b - 3c + d = -48
-27 + 9b - 3*(-1) +d = -48 {From (I)}
-27 + 9b + 3 + d = - 48
9b + d = -48 + 27 - 3
9b + d = -24 --------------(III)
Subtract equation (III) from equation (II),
(II) 4b + d = -9
(III) 9b + d = -24
- - +
-5b = 15
b = 15 ÷ (-5)
[tex]\boxed{b=-3}[/tex]
Plugin b = -3 in equation (II),
4*(-3) + d = -9
-12 + d = -9
d = -9 + 12
[tex]\boxed{d = 3}[/tex]
[tex]\boxed{\bf f(x) = x^3 - 3x^2 -x + 3 }[/tex]
please help asap im begging
Answer:
The diameter of the circle is 12 inches, which means the radius of the circle is half of the diameter, or 6 inches.
The formula for the area of a circle is:
A = πr^2
where A is the area of the circle and r is the radius.
Plugging in the value for the radius, we get:
A = π(6 in)^2
A = π(36 in^2)
A = 113.1 in^2 (rounded to one decimal place)
Therefore, the area of the circle is approximately 113.1 square inches.
Harold will drive 330 more miles at the same rate how long will it take care old to drive 330 miles
Assuming he drives at a speed of 60 miles per hour, it will take him 5.5 hours to drive the 330 miles.
How long does it takes him to drive the 330 miles?If Harold drives 330 more miles at the same rate of 60 miles per hour, we can use the formula:
distance = rate x time
Let t be the time in hours it takes to drive the additional 330 miles. Then we have:
330 = 60t
Solving for t, we get:
t = 330/60
t ≈ 5.5 hours
Therefore, it takes approximately 5.5 hours to drive the additional 330 miles at a rate of 60 miles per hour.
Full question "Harold will drive 330 more miles at the same rate. Assume he drives at a speed of 60 miles per hour, how long does it take to drive the 330 miles?'
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Find m∠AZJ using the figure below
In Linear pair, 52° is the value ∠AZJ .
What is Linear pairs?
When two lines meet at a single point, a pair of linear angles is created. If, following the junction of the two lines, the angles are next to one another, they are said to be linear. A linear pair's total angles are always equal to 180 degrees.
A pair of adjacent angles is said to be a linear pair if the total of the two angles' (measured) angles is 180°. This means that a linear pair of angles always adds up to 180°. The linear pair of 30°, for instance, is 150°; the linear pair of 70°, 110°; etc.
∠JZF or ∠HZG
Linear pairs form supplementary angles.
∠AZJ = ∠AZH - ∠HZA
= 90° - 38°
∠AZJ = 52°
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A town has a population of 12,000 and grows at 3. 5% every year. What will be the population after 7 years, to the nearest whole number?
If the population growth rate is 3.5 percent every year then the population of the town after 7 years would be 14940.
Given that population grows 3.5 percent every year.
So, the increase in population after one year
= 3.5% of 12000
= (3.5/100) × 12000
= 420
Thus the increase in population after 7 year would be,
= population increase in one year × 7
= 420×7 = 2940
Hence population of the town after 7 years = (present population + increase in population)
= 12000 + 2940
= 14940
So the population of the town after 7 years with 3.5 % growth every year would be 14490.
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Jessica went deep sea diving. She make the first stop on her descent at 25 meters below the surface of the water. From that point she dives down further, stopping every 5 meters. If she makes 4 additional stops, which number represents her position, relative to the surface of the water?
*
A 45
B 20
C -20
D -45
Answer:
-45
Step-by-step explanation:
each letter of the alphabet is written on a piece of paper and put in a hat. if a piece of paper is chosen at random, what is the probability that it will be a vowel? the number of favorable outcomes is . the number of possible outcomes is . so p(vowel)
The probability of selecting a vowel from a hat with letters of the alphabet is 5/26 or approximately 0.1923. There are 5 vowels and 26 total letters.
There are 5 vowels in the English alphabet: A, E, I, O, and U. Therefore, the number of favorable outcomes (i.e., the number of vowels in the hat) is 5.
There are a total of 26 letters in the English alphabet. Therefore, the number of possible outcomes (i.e., the number of letters in the hat) is 26.
The probability of selecting a vowel is the ratio of the number of favorable outcomes to the number of possible outcomes:
P(vowel) = number of favorable outcomes / number of possible outcomes
P(vowel) = 5 / 26
Therefore, the probability of selecting a vowel is 5/26 or approximately 0.1923 (rounded to four decimal places).
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Answer:
5
26
5/26
19
Step-by-step explanation:
10 ft
G
Find the area.
8 ft
Remember: A = r²
A = [?] ft²
Round to the nearest
hundredth.
Use 3.14 for T.
The area of the figure composing of a rectangle and a semi-circle is 105.12 ft².
What is the area of the composite figure?The figure in the image composes of a rectangle and a semi-circle.
The area of a rectangle is expressed as;
Area = Length × Width
The area of a semi-circle is expressed as;
Area = 1/2 × πr²
First, we find the area of the rectangle.
Area = Length × Width
Plug in Length = 10ft and Width = 8ft
Area = 10ft × 8ft
Area = 80ft²
Next, find the area of the semi-circle.
Area = 1/2 × πr²
Since diameter = 8ft, radius r = diameter/2 = 8ft/2 = 4ft
Area = 1/2 × 3.14 × (4ft)²
Area = 1/2 × 3.14 × 16 ft²
Area = 25.12 ft²
Now, area of the composite figure will be:
Area of the rectangle + Area of the semi-circle.
Area = 80ft² + 25.12 ft²
Area = 105.12 ft²
Therefore, the area of the figure is 105.12 ft².
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A company ordered 20 boxed lunches from a deli for $120.40. Each boxed lunch cost the same amount. Write an equation that represents the proportional relationship between y, the total cost of the boxed lunches, and x, the number of boxed lunches.
Answer:
Step-by-step explanation:well work it out use ur brain
Answer:
Step-by-step explanation:you need to multiply 120x40=20 your get $2,408