An architect designs a rectangular flower garden such that the width is exactly two-thirds of the length. If 400 feet of antique picket fencing are to be used to enclose the garden, find
the dimensions of the garden.

Answers in bold

====================================================

Explanation:

Always start the proof with what you are given. It seems silly to repeat the given information, but it's just how all proofs are formed. This is to build from those given statements to lead to what we want to prove on the last line.

The term "midpoint" refers to splitting a segment into two equal smaller pieces. So for instance, if Q is the midpoint of PR, then the pieces PQ and QR are equal. Each smaller piece is half as long as PR.

The transitive property is the idea if A = B and B = C, then A = C. Think of it like substitution. Or you can think of a set of dominoes: A knocks down B, which knocks down C. Therefore A knocks down C.

The addition property of equality is where we go from A = B to A+C = B+C. I've added C to both sides. Adding the same thing to both sides will not change the equation. It keeps things balanced. When going from line 4 to line 5, we added PQ to both sides.

In line 6, we've gone from PQ+PQ to 2PQ.

In line 7, we replace PQ with QR on the right hand side. We use the substitution property here (refer to line 3 where we found PQ = QR)

Line 8 uses the segment addition postulate. If A,B,C are collinear, then AB+BC = AC. The point B doesn't necessarily need to be the midpoint.

Line 9 uses the transitive property again (we combine lines 7 and 8 to lead to line 9).

The final line has us divide both sides by 2 to get what is shown in the table. This is the division property of equality.

------------------------

Shortcut using informal logic:

PQ = QR

QR is half as long as QS

Therefore, PQ is also half as long as QS

Answer:

The last reason for  [tex]PQ=\frac{1}{2}QS[/tex]  is division property of equality

Step-by-step explanation:

The reason for  [tex]PQ=\frac{1}{2}QS[/tex]  can be evaluated as follows    

For the statement [tex]Q[/tex] is the midpoint of [tex]\overline{PR}[/tex], the reason is given

For the statement [tex]R[/tex] is the midpoint of [tex]\overline{QS}[/tex], the reason is given

For the statement [tex]PQ=QR, QR=RS[/tex] , the reason is definition of midpoint

The midpoint of a line segment is the point that separates the line segment into two congruent parts.

For the statement [tex]PQ=RS[/tex] , the reason is transitive property

Transitive property says that if [tex]a=b[/tex], and [tex]b=c[/tex], then [tex]a=c[/tex]

In this statement [tex]PQ=QR, QR=RS[/tex], then [tex]PQ=RS[/tex]

For the statement [tex]PQ+PQ=PQ+RS[/tex] , the reason is addition property of equality

Addition property of equality says that if [tex]a=b[/tex], then [tex]a+c=b+c[/tex]

For the statement [tex]2PQ=PQ+RS[/tex] , the reason is combining of like terms of algebraic expression

Combining like terms is the process to add or to substract the coefficient of the terms

In this statement combining like terms is at the left side of the equation,  [tex]PQ+PQ=2PQ[/tex]

For the statement [tex]2PQ=QR+RS[/tex] , the reason is substitution property

Substitution property of equality says that if [tex]a=b[/tex], then [tex]a[/tex] replaces [tex]b[/tex] in any equation, in this statement [tex]QR[/tex] replaces [tex]PQ[/tex]

For the statement [tex]QR+RS=QS[/tex] , the reason is substitution property and definition of midpoint

In this statement [tex]QR+RS[/tex] replaces [tex]2PQ[/tex], and [tex]QS[/tex]  is the sum of two congruent parts  [tex]QR+RS[/tex]

For the statement [tex]2PQ=QS[/tex] , the reason is substitution property

In this statement [tex]2PQ[/tex] replaces [tex]QR+RS[/tex]

For the statement  [tex]PQ=\frac{1}{2}QS[/tex]  , the reason is division property of equality

Division property of equality says that if [tex]a=b[/tex], and [tex]c\neq 0[/tex], then [tex]\frac{a}{c} =\frac{b}{c}[/tex]

In this statement [tex]\frac{2PQ}{2} =\frac{QS}{2}[/tex] or can be reduced into [tex]PQ=\frac{1}{2}QS[/tex]

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The trigonometric ratio (trig ratio) for the angle in this right-angled triangle should be expressed as fractions in simplest terms as follows:

Sin(M) = 21/121

Cos(L) = 10/11

Tan(K) = 21/100

In Mathematics, a trigonometric ratio (trig ratio) can be calculated by using this mnemonic SOHCAHTOA:

Sinθ = Opposite/Hypotenuse

Sin(M) = KM/ML

Sin(M) = √21/11

Taking the square of both the numerator and denominator, we have:

Sin(M) = (√21)²/11²

Sin(M) = 21/121

For the cosine of an angle in a right-angled triangle, we have:

Cosθ = Adjacent/Hypotenuse

Cos(L) = KL/ML

Cos(L) = 10/11

For the tangent of an angle in a right-angled triangle, we have:

Tanθ = Opposite/Adjacent

Tan(K) = KM/KL

Tan(K) = √21/10

Taking the square of both the numerator and denominator, we have:

Tan(K) = (√21)²/10²

Tan(K) = 21/100

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The height of the arrow at 2.5 seconds is 71 feet.

Given:

An arrow released from a bow, traveling towards a target 200 yards away,reaches a max height of 42.6 ft in only 1.5 seconds.

1.5 seconds = 42.6 feet

divide by 1.5 on both sides.

1 sec = 28.4 feet

0.5 sec = 28.4/2

0.5 sec = 14.2 feet

2.5 sec = 1.5 sec + 1 sec

= 42.6 + 28.4

= 71 feet

Therefore The height of the arrow at 2.5 seconds is 71 feet.

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Answer:

1,148

Step-by-step explanation:

Answer:Marsha earned interest = $850

Step-by-step explanation:

Amount deposited = $8500

Time = 2 years

Interest rate = 5% = 0.05

We need to find how much did Marsha earn in interest

The formula use is:

Where I is interest earned, P is amount deposited, r is interest rate and t is time.

Putting values and finding interest.

So, Marsha earned interest = $850

The error is that the student used the formula for a reflection in the x-axis, and the correct endpoints are: C'(1, 1) and D'(3, -2)

Rotation simply means turning.

In a cartesian plane, if a point P, whose position vector is (x,y), is rotated through 270° about the origin, the position of the image P' is (y, -x)

But the reflection in the x-axis is defined as (x,y) => (x, -y) i.e. if a position vector of a point is (x,y), the position vector of its image in the x-axis is  (x, -y)

Given: C(-1, 1) and D'(2, 3)

Therefore, the student used the formula for a reflection in the x-axis. The 2nd option is the answer.

The correct endpoints are: C'(1, 1) and D'(3, -2)

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For the given composite function, then values of

(a) 63x + 8

(b) 63x + 56

(c) 81x + 80

(d) 49x

Composite functions:

An operation where two functions say f and g generate a new function say h in such a way that h(x) = g(f(x)) is known as composite functions.

Given,

Here we have the functions f(x) = 9x + 8 and g(x) = 7x

Now, we have to find the composite function of each.

(a) The composite function is written as,

=> f(g(x)) =f(7x)

=> f(7x) = 9(7x) + 8

=> f(g(x)) = 63x+ 8

(b) The composite function is written as,

=> g(f(x))

=> g(f(x)) =g(9x + 8)

=> g(9x + 8) = 7(9x + 8)

=> g(9x + 8) = 63x + 56

(c) The composite function is written as,

=>  f(f(x))

=> f(9x +8) = 9(9x + 8) + 8

=> f(9x + 8) = 81x + 72 +8

=> f(9x + 8) = 81x + 80

(d) The composite function is written as,

=> g(g(x))

=> g(7x) = 7(7x)

=> g(7x) = 49x

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Part a; compounded semiannually (n = 2); CI =  $6,390.

Part b: compounded quarterly ( n = 4) ; CI =  $6,425

Part c: compounded monthly (n = 12): CI =  $6,450

Part d: compounded continuously; CI =  $6,467.

The compound interest is calumniated by formula.

A = P(1 + r/100n)∧nt

In which,

A  = amount after compounding

P = principal amount  ($20,000 )

n = number of time principal amount compounded in a year.

t = total time in years (7 years)

r= rate of interest (4%)

Part a; compounded semiannually (n = 2)

A = P(1 + r/100n)∧nt

A = 20,000(1 + 4/200)∧(2x7)

A = 26390

CI = A - P

CI =  26,390 -  20,000

CI =  $6,390.

Part b: compounded quarterly ( n = 4)

A = 20,000(1 + 4/400)∧(4x7)

A = 26425

CI = A - P

CI =  26,425 -  20,000

CI =  $6,425

Part c: compounded monthly (n = 12)

A = 20,000(1 + 4/1200)∧(12x7)

A = 26450

CI = A - P

CI =  26,450-  20,000

CI =  $6,450

Part d: compounded continuously.

P = P₀e∧rt

P = (20,000)(2.72)∧(0.04 x 7)

P = 20,000 x 1.32

P = 26,467

CI =  26,467 - 20, 000

CI =  $6,467.

Thus, the for compounded continuously is $6,467.

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Answer:

x=47

Step-by-step explanation:

Sum of all three angles of a triangle is 180 degrees

We know one angle which is 51 degrees

The second is 73 degrees since it’s corresponding angle is 73 degrees

73+51=124

Now subtract this from 180

180-124=56

We can set x+9 equal to this

X+9=56

  -9.  -9

x=47

Hopes this helps please mark brainliest

The distance formula used to determine if the figure is a parallelogram indicates;

A parallelogram is a quadrilateral that have two pairs of facing sides that are parallel and of the same length.

The length of the sides can be found by using the formula for finding the distance between two points on the coordinate plane as follows;

The distance between the points (x₁, y₁) and (x₂, y₂) is d = √(x₂ - x₁)² + (y₂ - y₁)²

The length of QR = √((1 - (-10))² + (-1 - (-2))²) = √(122) ≈ 11.0

Length of ST = √((1 - (-11))² + (-7 - (-8))²) = √(145) ≈ 12.0

Length of QS = √((1 - (-10))² + (-7 - (-2))²) = √(146) ≈ 12.1

Length of RT = √((1 - (-11))² + (-1 - (-8))²) = √(193) ≈ 13.9

The lengths of the opposite sides of a parallelogram are the same. The lengths of the sides of the quadrilateral QRST are different, therefore, quadrilateral QRST is not a parallelogram.

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Answer:7/10

Step-by-step explanation:

Answer:

0

Step-by-step explanation:

The y-intercept and x-intercept of the following linear equation is

x intercept

x intercept refers to the value of x when y is zero

−32x−3y=3

at y = 0

-32x - 3 * 0 = 3

-32x = 3

x = -3/32 = -0.09375

the coordinate is (-0.094, 0)

y intercept

y intercept refers to the value of y when x is zero

−32x −3y=3

at x = 0

0 - 3y = 3

y = -1

the coordinate is (0, -1)

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The area of this triangle is 21 square units.

From the graph:

The length of the base = 4 + 3 units

= 7 units

The length of the height = 3 + 3

= 6 units

Area of the triangle = 1/2 b *h

= 1/2*7*6

= 1*42/2

= 42/2

= 21 square units.

Therefore the area of this triangle is 21 square units.

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Order of the tiles D, C, E, A

A square root of a number is a value of a number(other than the original number) that when multiplied by itself gives the original number. A square root of a number will be in the form of a².

Here are some examples of the square root of a number

√4 = 2 [ since 2 × 2 = 4 ]

√9 = 3 [ since 3 × 3 = 9 ]  

Here we have

1. √90 - √40

2. (√35 - √42) / (√5 - √6)

3. 2√27 - √48  

4. (√54 -√24)/√18 - √8  

Each problem can be solved as given below

1. √90 - √40

= √9 × 10 - √4 × 10  

= 3√10 - 2√10

= √10 = 3.16227

2.  (√35 - √42) / (√5 - √6)  

= (√5 × √7 - √7 × √6 ) / (√5 - √6)  

= √7 [√5 - √6 ] / (√5 - √6)  

= √7  = 2.64575

3. 2√27 - √48  

= 2√9×3 - √16×3

= 6 √3 - 4√3

= 2√3 =  3.46410

4. (√54 -√24)/√18 - √8  

= (√9×6 - √4×6)/√9×2 - √4×2 )

= (3√6 - 2√6)/3√2 - 2√2 )  

= (√6) / √2 )

= 1.73205

The resultant decimal numbers are

3.16227, 2.64575, 3.46410, and 1.73205.

And the order of the decimal numbers is

1.73205 < 2.64575 < 3.16227 < 3.46410

Therefore,

Order of the tiles D, C, E, A

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By applying the exterior angle property, the measure of the exterior angle to ∠G in triangle EFG is equal to 120.7°

In Mathematics, the exterior angle property can be defined as a theorem which states that the measure of an exterior angle in a triangle is equal in magnitude to the sum of the measures of the two remote or opposite interior angles of that triangle.

In order to determine the value of ∠G in triangle EFG (ΔEFG), we would have to apply exterior angle property. Mathematically, the exterior angle property can be used to model triangle EFG (ΔEFG) and this is given by:

m<G = m<E + m<F

Substituting the given parameters into the formula, we have;

m<G = 84.3° + 36.4°.

m<G = 120.7°.

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Answer:

0.559%

Step-by-step explanation:

The chance of winning equals the number of tickets bought divided by the total number of tickets

2 tickets are bought by the person, and 358 tickets are sold in total

2/358 would equal 0.00558659218..., and the percentage would be 0.558659...%
Rounding 0.558659 to the nearest thousandth's place would yield
0.559, so 0.559% would be the final answer

3.5m + 2.5 = 79.5 is the equation that could be used to determine m, the number of miles in the taxi ride.

Given in question,

Pick-up fee = $2.50

Fee per mile = $3.50

Total fare = $79.50

Let the no. of miles = m

Fee for m miles = 3.50m

Equation, 3.50m + 2.50 = 79.50.

Hence, the equation is 3.50m + 2.50 = 79.50.

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Answer:

[tex]\frac{2x\sqrt{3y}}{{3y}}[/tex]

Step-by-step explanation:

Apply fractional law for squares:

[tex]\sqrt{\frac{4x^2}{3y}} = \frac{\sqrt{4x^2}}{\sqrt{{3y}}}[/tex]

Then simply:

[tex]\frac{2x}{\sqrt{{3y}}}[/tex]

Rationalize denominator:

[tex]\frac{2x}{\sqrt{{3y}}} \cdot \frac{\sqrt{{3y}}}{\sqrt{{3y}}} = \frac{2x\sqrt{3y}}{{3y}}[/tex]

Answer:25/.2

Step-by-step explanation:

i think this is right

Answer:

The answer would be 4 7/24

Step-by-step explanation:

We can obtain common denominators by multiplying both numerator (top) and denominator (bottom) by the same amount.

The LCM (least common multiple) for your question would be 24, so you would multiply in order to get the fractions to have a common denominator of 24, while keeping the same value.

The length of the rectangle is 56.9 unit and the breadth of the rectangle is 25.9 unit.

Area of the rectangle is 1473.7 sq. unit.

Given, a rectangle's length is 5.1 more than 2 times its width.

Its perimeter is 165.6.

Let the length of the rectangle be l,

breadth of the rectangle be b.

According to the question,

l = 5.1 + 2b

Using the formula of Perimeter,

Perimeter = 2(l + b)

165.6 = 2(5.1 + 2b + b)

165.6 = 2(5.1 + 3b)

82.8 = 5.1 + 3b

3b = 77.7

b = 25.9

Now, l = 5.1 + 2(25.9)

l = 5.1 + 51.8

l = 56.9

So, the length of the rectangle is 56.9 unit

and the breadth of the rectangle is 25.9 unit

Now, the area of the rectangle be,

Area = l×b

Area = 56.9×25.9

Area = 1,473.71

Area of the rectangle = 1473.7 sq. unit

Hence, the length of the rectangle is 56.9 unit and the breadth of the rectangle is 25.9 unit.

Area of the rectangle is 1473.7 sq. unit.

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The amount that the patient is receiving, in mcg/min, using proportions, is of:

1008 mcg/min.

A proportion represents a unit rate relative to a total amount, and this unit rate is used along basic arithmetic operations, especially multiplication and division, to find the desired measures in the context of a problem.

Considering than an hour has 60 minutes, the amount infused each minute is given as follows:

40.5/60 = 0.675 mL/min.

The rate of milligrams per milliliters is given as follows:

400/250 = 1.6 mg per mL.

Hence the number of mg received per minute is:

1.6 x 0.675 = 1.08 mg per minute.

1 mg equals 1000 mcg, hence the number of mcg received per minute is:

1.08 x 1000 = 1008 mcg/min.

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Answer:

The solution is 4 7/8 or 39/8

Step-by-step explanation:

[tex] \sf 3 \frac{5}{8} + 1 \frac{2}{8} \\ \\ = \sf (3 + 1) + ( \frac{5}{8} + \frac{2}{8} ) \\ \\ = \sf 4 + \frac{7}{8} \\ \\ = \sf 4 \frac{7}{8} [/tex]

Or

[tex]\sf 3 \frac{5}{8} + 1 \frac{2}{8} \\ \\ = \frac{29}{8} + \frac{10}{8} \\ \\ = \sf \frac{39}{8} [/tex]

Answer:

75% - 20% = 55%

Step-by-step explanation:

From the question, we could give annotations for the number of tickets sold and the amount of money in the cashbox. We also find that there is 2 other elements related to the relationship between the tickets sold and the amount of money in the cashbox; the price of each ticket and the amount of money in the cashbox before any tickets are sold. Hence we could make some annotations:

m = money in the cashbox (after tickets are sold)

p = price of each tickets

x = number of tickets sold

c = money in the cashbox (before tickets are sold)

Based on the question, we know that the amount of money in the cashbox is dependeding on the number of tickets sold and the original amount of money in the cashbox before any tickets are sold. The relationship between the money in the cashbox and the number of tickets sold is depending on the price of each ticket. Based on this understanding, we could rewrite the relationship into such equation:

m = px + c ... (i)

Using data from the question, we could input the datas into equation (i):

m = px + c

320 = 20p + c ... (ii)

492.50 = 43 p + c ... (iii)

980 = 108 p + c ... (iv)

By using equation (ii) and (iv) we could find the value of p

320 = 20p + c

980 = 108p + c -

660 = 88p

p = 7.5 ... (v)

Using the finding value of p, we could find the value of c by using subtitution of equation (v) into equation (ii):

320 = 20p + c

320 = 20(7.5) + c

320 = 150 + c

c = 170 ... (vi)

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Answer:

  (a)  68 grams

  (b)  42.5%

Step-by-step explanation:

You mix 100 g of 35% copper with 60 g of 55% copper to form an alloy. You want to know the number of grams and the percentage of copper in the mixture.

The total amount of copper is ...

  0.35·(100 g) +0.55·(60 g) = 35 g + 33 g = 68 g

There are 68 grams of copper in the resulting mixture.

The percentage of copper in the mixture is ...

  (68 g)/(100 g + 60 g) × 100% = 68/160 × 100% = 42.5%

The resulting mixture is 42.5% copper.

The probability that exactly 4 insects will survive is 0.8975

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, 0 indicates impossibility of the event and 1 indicates certainty.

probability= sample space/total possible outcome

the total possible outcome for the insecticide to kill insect is 60%

If 4 will survive out of 13, it means 7 will be killed by the insecticide, the percentage is now 7/13×100 = 53.85%

Therefore the probability for 4 insect to survive is 53.85/60= 0.8975

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Answer:

Okay, so let's make the variable x the 60% juice.

The equation:

0.60(x) + 1(120-x) = 0.7(120)

Multiply both sides by ten for easier calculation:

6x + 1200 - 10x = 840

Put x on one side and the numbers on the other:

-4x = -360

the negatives cancel out...

4x = 360

define x

x = 90

Step-by-step explanation:

The expression we get after measuring prq is 5610x²+ 891x+33.

Given that,

p is 33, r is 17x+1 and q is 10x+1

We have to find the measure of prq.

Take the,

p is 33, r is 17x+1 and q is 10x+1

Now,

prq

By multiply we get an expression

Multiplying each term to each other.

(33)(17x+1) (10x+1)

1st multiply 33 to 17x+1

(561x+33) (10x+1)

Now multiply each 561x with 10x+1 and 33 with 10x+1

5610x²+ 561x+ 330x+33

Adding 561x and 330x

5610x²+ 891x+33

Therefore, The expression we get after measuring prq is 5610x²+ 891x+33.

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