Proved
Used identities
cos (a - b) = cos a cos b + sin a sin b,sin²a + cos²a = 1,(a + b)² = a² + 2ab + b²Pls help answer fast rapidly is of khan academy:
Answer:
expanded: 7+0.08
standard: 7.08
Step-by-step explanation:
The hundredths place is the second digit after the decimal, so 8 hundredths is 0.08.
A group of 23 adults and 45 children are attending a play. Each row in the theater can seat 9 people.
What is the least number of rows needed to seat the group?
Answer:
At least 8 rows
Step-by-step explanation:
45 divided by 9 is 5. That is 5 rows
9x3= 27 and there are 23 adults. So that is 3 rows
5+3=8
8 rows in total.
Write this ratio as a fraction in simplest form without any units.
Answer:
change 5 weeks to days(5*7)-35days
then convert the days to seconds
35*86400/10*86400
answer 7/2
tic is an isosceles triangle with a vertex angle i if the measure of i90
Applying the definition of an Isosceles triangle, given that isosceles triangle TIC has a vertex angle I of 90 degrees, the measure of angle T is 45 degrees.
In geometry, an isosceles triangle is a triangle that has at least two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.
The two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs and base. Every isosceles triangle has an axis of symmetry along the perpendicular bisector of its base. The two angles opposite the legs are equal and are always acute, so the classification of the triangle as acute, right, or obtuse depends only on the angle between its two legs.
An Isosceles triangle has a vertex angle and two equal base angles.
Given that triangle TIC is an isosceles triangle:
angle I is the vertex angle = 90 degrees
angle T is a base angle
angle C is also a base angle
Thus:
angle C and angle T are congruent or equal to each other.
Therefore,
Angle T = (180 - 90)/2
⇒ Angle T = 90/2
⇒ Angle T = 45 degrees.
applying the definition of an isosceles triangle, given that isosceles triangle TIC has a vertex angle I of 90 degrees, the measure of angle T is 45 degrees.
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A line with a slope of 1 passes through the point (5, 3). What is its equation in slope "-intercept" form?
Answer: y=x−2 y = x − 2
Step-by-step explanation:
What are the 4 types of properties?
There are four basic properties of numbers: commutative, associative, distributive, and identity.
1.Commutative Property
a. Addition
. When two numbers are added, the sum is the same regardless of the order in which the numbers are added.
3 + 5 = 8 or 5 + 3 = 8
b. Multiplication. When two numbers are multiplied together, the product is the same regardless of the order in which the numbers are multiplied.
3 x 5 = 15 or 5 x 3 = 15
2.Associative Property
a. Addition. When three or more numbers are added, the sum is the same regardless of the way in which the numbers are grouped.
6 + (4 + 3) = 13 or (6 + 4) + 3 = 13
b. Multiplication. When three or more numbers are multiplied, the product is the same regardless of the way in which the numbers are grouped.
6 x (4 x 3) = 72 or (6 x 4) x 3 = 72
3.Distributive Property
The sum of two numbers times a third number is equal to the sum of each addend times the third number.
5 x (7 + 2) = 45 or 5 x 7 + 5 x 2 = 45
4.Identity Property
a. Addition. The sum of any number and zero is that number.
12 + 0 = 12
b. Multiplication, The product of any number and one is that number.
18 x 1 = 18
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david performed the following mathematical operation
1 must be the root of the polynomial [tex]$3 x^2+x-4$[/tex].
We are given that polynomial [tex]$3 x^2+x-4$[/tex] is being divided by factor [tex]$(\mathrm{x}-1)$[/tex]
It is also said that when David perform the mathematical operation it gives remainder 0 .
Remainder 0 represents that divisor [tex]$(\mathrm{x}-1)$[/tex] is a factor polynomial [tex]$3 x^2+x-4$[/tex].
As x-1 is a factor of the given ploynomial, we can get one of the root of by setting it equal to 0 and solve for x.
Therefore,
x-1=0
Adding 1 on both sides, we get
[tex]$$\begin{aligned}& x-1+1=0+1 \\& x=1 .\end{aligned}$$[/tex]
Therefore, 1 must be the root of the polynomial [tex]$3 x^2+x-4$[/tex].
Polynomials are sums of terms of the form k⋅xⁿ, where k is any number and n is a positive integer. For example, 3x+2x-5 is a polynomial. Introduction to polynomials. This video covers common terminology like terms, degree, standard form, monomial, binomial and trinomial.
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An airplane is flying 700 feet above ground. From the plane to the base of the control tower, the angle of depression is 47◦ . How far away is ground directly underneath the plane to the control tower? Round your answer to the nearest tenth.
Using trigonometric ratio the distance to the control tower is 750.7ft
Trigonometric RatioIn trigonometry, there are six trigonometric ratios, namely, sine, cosine, tangent, secant, cosecant, and cotangent. These ratios are written as sin, cos, tan, sec, cosec(or csc), and cot in short.
The values of these trigonometric ratios can be calculated using the measure of an acute angle, θ in the right-angled triangle given below. This implies that the value of the ratio of any two sides of the triangle here depends on angle C.
Using trigonometric ratio, the distance is
tan θ = opposite / adjacent
tan 47 = x / 700
x = 700 tan 47
x = 750.7 ft
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Evaluate the expression for j = –5, k = 2, and m = 3.
jk2 − m = ?
Answer: -23
Step-by-step explanation:
[tex]jk^2 -m=(-5)(2^2)-3=-5(4)-3=-20-3=-23[/tex]
How do you find the degree of a polynomial in precalculus?
Answer:
add up the exponents of each term and select the highest sum.
A college student has budgeted $25 a month for phone services. If the telephone
company charges $17 a month for basic service and $0.05 a minute for each long
distance call, how many minutes can be spent on long distance calls each month?
(Define a variable, create an equation, solve using algebra, and answer in a sentence.)
Answer:
160 minutes can be spent on long distance calls each month.
Step-by-step explanation:
first we know that there is standard charge of $17 for basic service, $0.05 cost per minute with a budget of $25.
let's write that into an equation to solve for (a) which will be number of minutes.
17 + a(0.05) = 25
solving:
17 + a(0.05) = 25
carry across the 17 whilst changing the sign, or some people is also thought to minus on both sides to get rid of a certain variable.
a(0.05) = 25 - 17
a(0.05) = 8
carry across the multiplication of 0.05 which will then turn into division, or divide by 0.05 on both sides to remove it.
[tex]a = \frac{8}{0.05} [/tex]
a = 160
therefore 160 minutes can be spent on long distance calls each month.
You can redefine your answer more simply to hours by dividing by 60, which gives 2 hours and 40 minutes
What triangle has 2 congruent sides and 2 congruent angles?
If triangle has 2 congruent sides and 2 congruent angles ,such a triangle is known as Isosceles triangle.
What is congruent triangle?
Congruence of triangles: 2 triangles square {measure} aforesaid to be congruent if all 3 corresponding sides square measure equal and every one the 3 corresponding angles square measure equal in measure. These triangles may be slides, rotated, flipped and turned to be looked identical. If repositioned, they coincide with one another. The image of congruity is’ ≅’.
Main Body:
A triangle with two congruent sides and congruent base angles is termed as an isosceles triangle.
If two sides of a triangle are congruent, then by isosceles triangle property, the angles opposite to these sides are congruent.
That is, base angles of the triangle are congruent.
Then, such a triangle is known as Isosceles triangle.
Therefore, option B is correct.
The actual question with option is in figure.
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How do you prove a trapezoid is isosceles?
Answer:
A trapezoid is isosceles if and only if its diagonals are congruent. So if we can prove that the bases are parallel and the diagonals are congruent, then we know the quadrilateral is an isosceles trapezoid, as Cool Math accurately states.
Step-by-step explanation:
Suppose that a cup of soup cooled from 90 degrees C to 60 degrees C after 25 minutes in a room whose temperature was 20 degrees C. Use Newton's law of cooling to answer the following questions.
(A) How much longer would it take the soup to cool to 30 degree C?
(B) Instead of being left to stand in the room, the cup of 90 degree C soup is put in the freezer whose temperature is - 10 degree C. How long will it take the soup to cool from 90 degree C to 30 degree C?
Answer:
A) 61.93 minutes
B) 40.93 minutes
Step-by-step explanation:
Newton's law of cooling
[tex]T(t)=T_S+(T_0-T_S)e^{-kt}[/tex]
where:
T(t) is the temperature of the object at time t.[tex]T_S[/tex] is the temperature of the surrounding environment.[tex]T_0[/tex] is the initial temperature of the object.t is time.k is a constant.Given values:
[tex]T_0[/tex] = 90°C[tex]T_S[/tex] = 20°Ct = 25 minutesT(t) = 60°CSubstitute the given values into the formula and solve for k:
[tex]\implies 60=20+(90-20)e^{-25k}[/tex]
[tex]\implies 40=70e^{-25k}[/tex]
[tex]\implies e^{-25k}=\dfrac{4}{7}[/tex]
[tex]\implies \ln e^{-25k}=\ln \dfrac{4}{7}[/tex]
[tex]\implies -25k \ln e=\ln \dfrac{4}{7}[/tex]
[tex]\implies -25k=\ln \dfrac{4}{7}[/tex]
[tex]\implies k=-\dfrac{1}{25}\ln \dfrac{4}{7}[/tex]
Part AGiven values:
[tex]T_0[/tex] = 60°C[tex]T_S[/tex] = 20°C[tex]k=-\dfrac{1}{25}\ln \dfrac{4}{7}[/tex]T(t) = 30°CSubstitute the given values into the formula and solve for t:
[tex]\implies 30=20+(60-20)e^{\frac{1}{25}t\ln \frac{4}{7}}[/tex]
[tex]\implies 10=40e^{\frac{1}{25}t\ln \frac{4}{7}}[/tex]
[tex]\implies e^{\frac{1}{25}t\ln \frac{4}{7}}=\dfrac{1}{4}[/tex]
[tex]\implies \ln e^{{\frac{1}{25}t\ln \frac{4}{7}}}=\ln \dfrac{1}{4}[/tex]
[tex]\implies {\dfrac{1}{25}t\ln \dfrac{4}{7}}\ln e=\ln \dfrac{1}{4}[/tex]
[tex]\implies {\dfrac{1}{25}t\ln \dfrac{4}{7}}=\ln \dfrac{1}{4}[/tex]
[tex]\implies t=\dfrac{\ln \dfrac{1}{4}}{{\frac{1}{25}\ln \frac{4}{7}}}[/tex]
[tex]\implies t=61.93063129...[/tex]
[tex]\implies t=61.93\;\; \sf minutes[/tex]
Part BGiven values:
[tex]T_0[/tex] = 90°C[tex]T_S[/tex] = -10°C[tex]k=-\dfrac{1}{25}\ln \dfrac{4}{7}[/tex]T(t) = 30°CSubstitute the given values into the formula and solve for t:
[tex]\implies 30=-10+(90-(-10))e^{\frac{1}{25}t\ln \frac{4}{7}}[/tex]
[tex]\implies 40=100e^{\frac{1}{25}t\ln \frac{4}{7}}[/tex]
[tex]\implies e^{\frac{1}{25}t\ln \frac{4}{7}}=0.4[/tex]
[tex]\implies \ln e^{\frac{1}{25}t\ln \frac{4}{7}}=\ln 0.4[/tex]
[tex]\implies{\dfrac{1}{25}t\ln \dfrac{4}{7}}\ln e=\ln 0.4[/tex]
[tex]\implies {\dfrac{1}{25}t\ln \dfrac{4}{7}}=\ln 0.4[/tex]
[tex]\implies t=\dfrac{\ln 0.4}{{\frac{1}{25}\ln \frac{4}{7}}}[/tex]
[tex]\implies t=40.93392072...[/tex]
[tex]\implies t=40.93\;\; \sf minutes[/tex]
What is the radius of the sphere given by x2 y2 z2 − 6y 8z 0?
The sphere's radius is 5, and its center is at (0, 3, -4).
What is a sphere?A sphere is a geometrical object that resembles a two-dimensional circle in three dimensions.
In three-dimensional space, a sphere is a collection of points that are all located at the same distance from a single point.
The radius of the sphere is denoted by the letter r, and the given point is its center.
List a few typical sphere examples from everyday life.
In everyday objects, sphere shapes can be found in the form of marbles, balls, oranges, yarn, and bubbles.
So, the center and radius of the sphere will be:
We have the sphere's equation, which is given by and has a radius of r.
(x - h)² + (y - k)² + (z - l)² = r² ...(1)
We must convert the following equation in order to determine the center and radius of the provided sphere.
x² + y² + z² - 6y + 8z = 0
By combining the x, y, and z terms and filling in the square, we obtain
x² + (y² - 6y + 9) +(z² + 8z + 16) = 9 + 16
(x - 0)² + (y - 3)² + (z + 4)² = 52
So, the sphere's center is at (0, 3, -4) and its radius is 5.
Therefore, the sphere's radius is 5, and its center is at (0, 3, -4).
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Complete question:
Find the center and radius of the sphere x2 + y2 + z2 - 6y + 8z = 0.
Solve for x
2 1/2x - 3/4 ( 2x + 5 ) = 3/8
Please explain step by step on how I get this answer to solve x!
Answer:
4 [tex]\frac{1}{8}[/tex]
Step-by-step explanation:
12. A factory needs to package 2,835 cartons
of orange juice in boxes. How many
more small boxes than large boxes are
needed to package the cartons of orange
juice?
Box Size
Small 15 number of cartons
Large. 35 number of cartons
The total number of cartons needed more to pack small 15 cartons than 35 large cartons is108 cartons.
What is the number of cartons needed?Division is defined as one of the four basic operations of arithmetic in mathematics.
The parameters for this question are:
Total number of cartons =2835
Number of small cartons needed = 15 cartons
Number of large cartons needed = 35 cartons
To package the orange juice in large cartons we need [tex]\frac{2835}{35} =81 cartons[/tex]
Also, to package 15 small cartons we need [tex]\frac{2835}{15} =189 cartons[/tex]
Number of more small cartons than large cartons is
Small cartons -large cartons
=189-81=108 more cartons.
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←
Diana has available 280 yards of fencing and wishes to enclose a rectangular area.
(a) Express the area A of the rectangle as a function of the width W of the rectangle.
(b) For what value of W is the area largest?
(c) What is the maximum area?
Answer:
(a) A = W(140 -W)
(b) W = 70
(c) 4900 square yards
Step-by-step explanation:
You want an expression for the rectangular area that can be enclosed by 280 yards of fence, if that area has width W. Further, you want the value of W that maximizes the area, and the size of that maximum area.
Perimeter expressionThe perimeter of the rectangle with length L and width W is ...
P = 2(L +W)
Then the length can be found from the width as ...
L = P/2 -W
For P=280, this is ...
L = 140 -W
(a) Area expressionThe area is the product of length and width:
A = LW
A = (140 -W)(W)
(b) Maximum areaThe area expression describes a downward-opening parabola with zeros at W=0 and W=140. The vertex (maximum) of the parabola is on the line of symmetry, halfway between these zeros. The value of W there is ...
W = (0 +140)/2 = 70
The area is largest for W = 70.
(c) AreaFor that value of W, the area is ...
A = (140 -70)(70) = 70² = 4900 . . . . . square yards
__
Additional comment
The generic solution to the area problem when fence is on all sides is that the shape is a square.
If the cost of fence is different for the different sides, the total cost in the east-west direction is equal to the total cost in the north-south direction when cost is minimized.
Joseph has $1.45 worth of nickels and dimes. He has a total of 17 nickels and dimes altogether. Determine the number of nickels and the number of dimes that Joseph has.
The number of nickels and the number of dimes that Joseph has are 12 and 5 respectively.
What is a linear equation?An equation of degree one is known as a linear equation.
A linear equation of two variables can be represented by ax + by = c.
Let the no. of nickels be N and no. of dimes be D.
∴ N + D = 17...(i) and 0.05N + 0.1D = 1.45...(ii)
Multiplying (ii) by 100 we get 5N + 10D = 145...(iii)
Now multiply (i) by 5 we get 5N + 5D = 85...(iv)
Now subtracting (iv) from (iii) we get,
5D = 60.
D = 12 hence N = 5.
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given the following measurements in parallelogram ABCD find the perimeter of CED
AC= 50 in AB= 18 in BD= 30in
The perimeter of triangle CED is 58 in
What is a Parallelogram ?
A unique variety of quadrilateral called a parallelogram has both pairs of its opposite sides parallel and equal.
Parallelograms' characteristics
(1) The opposing sides of a parallelogram are parallel to one another. In this case, AB || CD and AC || BD
(2) A parallelogram has equal-length opposing sides. In this case, AB = CD and AC = BD
(3) In a parallelogram, the measurements of the opposing angles are equal. In this case, A Equals C and B = D.
(4) The total of all the angles of a parallelogram, like all other quadrilaterals, is 360°.
(5) A parallelogram has neighboring or contiguous angles that sum to 180 degrees. As a result, A + B, B + C, C + D, and D + A all equal 180 degrees.
(6) A parallelogram's diagonals cut each other in half. In this case, OA = OC and OB = OD
(7) The parallelogram in the figure is split into two congruent triangles by the diagonals AC and BD.
The diagonals of a parallelogram bisect each other AC = 2 CE
Substitute AC= 50 into AC = 2CE = 50 = 2CE
Calculate 50 = 2 CE=> CE = 25
The diagonals of a parallelogram bisect each other BD = 2DE
Substitute BD= 30 into BD = 2DE => 30 = 2DE
Calculate 30 = 2DE => DE = 15
Opposite sides of parallelogram are congruent AB = CD
Substitute AB = 18 into AB = CD => CD = 18
Perimeter of a triangle PΔCDE = CE + CD + DE
Substitute CE = 25, DE = 15, CD = 18.
PΔCDE = CE + CD + DE
PΔCDE = 25+ 18 + 15
PΔCDE = 58 in
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CAN SOMEONE HELP WITH THIS QUESTION?✨
The equation that models the function is P(t) = 20400(1,09)⁺ and the fox population in 2008 is 40648
The equation of the functionFrom the question, we have the following parameters that can be used in our computation:
Initial population of fox = 20400
Rate of growth = 9 percent per year
These parameters can be presented as
Initial population of fox, a = 20400
Rate of growth, r = 9%
An exponential function is represented as
P(t) = a(1 + r)⁺
So, we have:
P(t) = 20400(1 + 9%)⁺
Evaluate
P(t) = 20400(1,09)⁺
The population in 2008Recall that
t = 0 in 2000
So, the value of t in 2008 is
t = 2008 - 2000
t = 8
Substitute t = 8 in P(t) = 20400(1,09)⁺
So, we have
P(8) = 20400(1.09)⁸
Evaluate
P(8) = 40648
So, the population is 40648
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is x^2+x+1=0 linear?
Answer:
No.
Step-by-step explanation:
Because the equation has an exponent, it is quadratic, and not linear.
Answer:1
Step-by-step explanation:
Mark needs to cut a board to use for a project. The board is 12 feet long. He first has to cut off 24 feet from one end of the board. He
then cuts the remaining section of board into 4 equal pieces.
Which equation can be used to determine the the length of each equal piece?
04r-2-12
02-4-12
O2+4=12
O 4x+2¹-12
Next
The equation that best defines the length of the equal piece is 4x+2¹-12
How to determine an equation?We should know that an equation is a mathematical model that shows the equality of two sides of an expression
Based on the given condition 12-24=4*x
Applying multiplicative distribution -12=4x
Rearrange the equation 4x=-12
Calculate the product of the quotient x=-3
Now 24feet divided by -3 =-4
Since it deals with length =4
In conclusion the equation that determines the length of each equal piece is 4x+2¹-12
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One integer is 2 times another. If the product of the two integers is 32, then find the integers
Answer:
The integers could be 8 and 4, just checked with a calculator
What are the types of inverse?
The various types of inverse function are the inverse of trigonometric functions, rational functions, hyperbolic functions and log functions
If the function is denoted by ‘f’ or ‘F’, then the inverse function is denoted by f⁻¹or F⁻¹.
A function accepts values, performs particular operations on these values and generates an output. The inverse function agrees with the resultant, operates and reaches back to the original function.
The various types of inverse function are:
Inverse Trigonometric Functions : The inverse trigonometric functions are also known as arc function as they produce the length of the arc.Inverse Hyperbolic Function : Just like inverse trigonometric functions, the inverse hyperbolic functions are the inverses of the hyperbolic functions. There are mainly 6 inverse hyperbolic functions exist which include sinh⁻¹ , cosh⁻¹, tanh⁻¹, csch⁻¹, coth⁻¹, and sech⁻¹Inverse Logarithmic Functions and Inverse Exponential Function : The natural log functions are inverse of the exponential functions.To know more about The inverse of function here
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Explain how you can use compatible numbers to estimate the quotient of
925÷29.
HELPP PLEASEE ITS 50 POINTS TYY
Answer:
We can be able to use compatible numbers to estimate the quotient of 925/29 if the whole number that is part of the decimal is less than the divisor, we need to be able to rename the decimal.
Step-by-step explanation:
Is a cubic graph exponential?
Yes, a cubic graph is an exponential.
What is Cubic Graph?
A cubic graph in the study of graph theory is a graph with all of its vertices having degree three. A cubic graph is a 3-regular graph, to put it another way. Trivalent graphs are another name for cubic graphs. A cubic bipartite graph is known as a bicubic graph.
What is Exponential?
The mathematical expression f(x)=exp or ex denotes the exponential function. Unless otherwise stated, the phrase usually refers to the positive-valued function of a real variable, though it can also be applied to complex numbers or extrapolated to other mathematical objects like matrices or Lie algebras.
Graphs of Exponential Functions if;
When the base is greater than 1, the exponential function has the following characteristics and you can easly understand from this :
The point is traversed by the graph (0,1)The domain only uses actual integers.It is between y>0.The graph is going up.The graph approaches negative infinity and becomes asymptotic to the x-axis.As x gets closer to positive infinity, the graph grows without bound.The chart is non-stop.The graph is rounded.Learn more about cubic graph exponential click here:
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A ball is dropped from a height of 10 ft and bounces 83% of its previous height on each bounce. How high off the ground is the ball at the top of the 4th bounce?
Answer:
The height of the 4th bounce would be 4.7458321; rounded would be 4.8.
Step-by-step explanation:
We can figure this question out by first converting the 83% into a decimal, by bringing back 2 decimal places:
0.83.
And then setting up this equation:
0.83 x 10 = 8.3
Now, this is for 1 bounce. We have to do this 4 more times:
0.83 x 8.3 = 6.889
3rd time:
0.83 x 6.889 = 5.71787
4th time:
0.83 x 5.71787 = 4.7458321.
Now, we can round this answer to:
4.8
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Nathan is building a toolshed with a rectangular floor. The area of a rectangle is given by the formula 1 w, where I represents
the length, and w represents the width. The floor of the shed will have measurements of either / = 9 and w =
= 7 or/= 8 and
w = 8. What is the area of the larger floor?
O63
O 64
O 56
O 72
The area of the larger floor is 64 square units when w = 8 and l = 8. which is the correct answer would be an option (B).
To compare the values, we must first calculate each given situation. The area of any given rectangle is length times width.
If w = 7 and l = 9
A = l × w = 8 × 6 = 48
If w = 8 and l = 8
A = l × w = 8 × 8 = 64
So, the second case would obtain a larger area.
Therefore, the area of the larger floor is 64 square units when w = 8 and l = 8.
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Determine whether the relation is a function
Explain why or why not.
1. (9, -1) (4,2) (5,3) (8, -1) (3, 0)
The relation:
(9, -1) (4,2) (5,3) (8, -1) (3, 0)
Is a function, because each element of the domain appears only once.
Is the relation a function?A relation maps elements from a set, called the domain, into elements from another set, called the range.
Such that the element x being mapped into the element y is written as:
(x, y).
A relation is a function only if each element in the domain is mapped into only one element of the range.
Here we have the relation:
{ (9, -1) (4,2) (5,3) (8, -1) (3, 0)}
We can see that no element of the domain appears twice, so yes, this is a function.
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