The length of a rectangle is 10 feet longer than it is wide. if each side is increased 10 feet, then the area is multiplied by 4. what was the size of the original rectangle?

To simplify the given expressions into one fraction, we find a common denominator for each set of fractions, adjust the numerators accordingly, and then combine the numerators over the common denominator.

To simplify the given expressions into one fraction, we need to find a common denominator and combine the fractions accordingly. Let's go through each expression step by step.

For the expression 7/x-3 + 3/x-5, the common denominator would be (x-3)(x-5). We need to multiply each fraction by the denominator that it's missing to get common denominators, and then sum the numerators over the common denominator.

The expression -5/x-3 - (-4)/(x+2) involves subtracting fractions. To simplify, we again find a common denominator, which is (x-3)(x+2), and proceed similarly to the first expression.

For 6/x+7 - 3/x-2, the common denominator is (x+7)(x-2). We perform the same process of equating denominators and combining.

To illustrate with the first expression:
(7(x-5))/((x-3)(x-5)) + (3(x-3))/((x-3)(x-5)) = (7x - 35 + 3x - 9)/((x-3)(x-5))

Combine the numerators to get a single fraction:

(10x - 44)/((x-3)(x-5))

Apply the same approach to the other two expressions to get them into a single fraction form.

Answer:

[tex]x=1\pm\sqrt{47}[/tex]

Step-by-step explanation:

We have been given an equation [tex]2x^2+3x-7=x^2+5x+39[/tex]. We are asked to find the solution for our given equation.

[tex]2x^2+3x-7=x^2+5x+39[/tex]

[tex]2x^2-x^2+3x-7=x^2-x^2+5x+39[/tex]

[tex]x^2+3x-7=5x+39[/tex]

[tex]x^2+3x-5x-7-39=5x-5x+39-39[/tex]

[tex]x^2-2x-46=0[/tex]

Using quadratic formula, we will get:

[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

[tex]x=\frac{-(-2)\pm\sqrt{(-2)^2-4(1)(-46)}}{2(1)}[/tex]

[tex]x=\frac{2\pm\sqrt{4+184}}{2}[/tex]

[tex]x=\frac{2\pm\sqrt{188}}{2}[/tex]

[tex]x=\frac{2\pm2\sqrt{47}}{2}[/tex]

[tex]x=1\pm\sqrt{47}[/tex]

Therefore, the solutions for our given equation are [tex]x=1\pm\sqrt{47}[/tex].

The missing number using Euler's formula is: Option A. 17

The maximum volume of a square pyramid is: Option B. 4,608

"It is a geometrical formula. V − E + F = 2, where V represents number of vertices, E represents number of edges and F represents number of faces."

"Square pyramid is a three dimensional geometrical figure where four triangular sides are associated to square base."

"A cube is a three-dimensional geometric structure with six congruent square face."

[tex]V=\frac{1}{3}a^{2}h[/tex]

where [tex]a[/tex] represents the length of square base and [tex]h[/tex] represents the height of the pyramid.

Consider the first question,

number of vertices (V) = 13

number of edges (E) = 28

So, using Euler's formula:

[tex]13-28+F=2[/tex]

⇒ [tex]-15+F=2[/tex]

⇒ [tex]F=2+15[/tex]

⇒ [tex]F=17[/tex]

So, the number of faces are 17.

Hence, the correct answer is option A. 17

Consider last question,

the side length of a cube = 24 cm

As the square pyramid fit inside a cube.

⇒ the length of the square base of a pyramid [tex]b[/tex] = 24 cm

and the height of a square pyramid [tex]h[/tex] = 24 cm

So, the volume of a square pyramid is,

[tex]V=\frac{1}{3} a^{2} h[/tex]

⇒ [tex]V=\frac{1}{3}[/tex] × [tex]24^{2}[/tex] × [tex]24[/tex]

⇒ [tex]V= 4608[/tex] [tex]cm^{3}[/tex]

Therefore, the maximum volume of a square pyramid that can fit inside a cube with a side length of 24 cm is [tex]4608[/tex] [tex]cm^{3}[/tex].

And the correct answer is option B. 4,608

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Answer: The center of this data is 2.

Step-by-step explanation:

Since we have given that

The data shows the number of Taylor's basketball team:

[tex]2, 0, 1, 2, 4, 1, 4, 0, 3, 2[/tex]

We need to find the center of this data.

As we know that "Median" gives the middle value of the data, So, it is known as "Center of this data".

1) First we write it in ascending order:

[tex]0,0,1,1,2,2,2,3,4,4[/tex]

2) Count the number of terms :

n=10

Since n is even.

3) As we know the formula for even number of data:

[tex]Me=\frac{\frac{n}{2}+({\frac{n}{2}+1)}}{2}\\\\Me=\frac{\frac{10}{2}+({\frac{10}{2}+)}}{2}\\\\Me=\frac{5^{th}+6^{th}}{2}\\\\Me=\frac{2+2}{2}\\\\Me=\frac{4}{2}\\\\Me=2[/tex]

Hence, The center of this data is 2.

Answer:

2

Step-by-step explanation:

2 is correct on plato

Answer: Age of Lin is 12

Solution:

Let X= age of Maya

(X/2)+4= age of Adrian

((X/2)+4)-7= age of Lin

X+(X/2)+4+((X/2)+4-7)=61

X+.5X+4+.5X+4-7=61

2X+4+4-7=61

2x=61-8+7

2X=60

X=30 age of Maya

19= age of Adrian

Age of Lin is

=((X/2)+4)-7

=15+4-7

=12

To check if this is correct

30+19+12=61

By setting up an algebraic equation to represent the relationship between the ages of Lin, Adrian, and Maya, and using the sum of their ages, we determined that Lin is 17 years old.

To solve this problem, let's use algebra to define the ages of Lin, Adrian, and Maya. Let's assume that Maya's age is X. Based on the information provided, Adrian is 4 years older than half of Maya's age, so Adrian's age is represented as (X/2) + 4. Lin is 7 years younger than Adrian, so Lin's age is (X/2) + 4 - 7, which simplifies to (X/2) - 3. The sum of the three ages is 61, so we can now set up an equation to find Maya's age and, subsequently, Lin's age.

The equation based on the su of their ages is:

X + (X/2) + 4 + (X/2) - 3 = 61

Combining like terms and solving for X:

2X + X + 8 - 6 = 122

3X + 2 = 122

3X = 120

X = 40

Now that we know Maya's age (X), we can find Lin's age:

(40/2) - 3 = 20 - 3 = 17

Therefore, Lin is 17 years old.

The correct equation to solve for x, given that angle LOM (measured as (x + 15)°) and angle MON (measured as 48°) are complementary, is (x + 15)° + 48° = 90°. Thus, the answer is option D.

The subject of this question is Mathematics, specifically it refers to geometry, solving for a variable, and understanding the concept of complementary angles. Let's analyze the options provided.

Two angles are said to be complementary if the sum of their measure is 90 degrees. So, if angle LOM and angle MON are complementary, the sum of m∠LOM and m∠MON should be 90°. Since the measure of m∠LOM is given as (x + 15)° and the measure of m∠MON is given as 48°, the equation that represents this relationship is (x + 15)° + 48° = 90°.

Therefore, option D is the correct choice to solve for x.

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Final answer:

To find the number N with LCM of 48 and GCF of 8 with 16, we use the formula LCM × GCF = N × 16 which gives N = 24.

Explanation:

To find the number N when given that it has a Least Common Multiple (LCM) of 48 with the number 16 and a Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD), of 8, we can use the relationship between LCM, GCF, and the product of the two numbers:

LCM(N, 16) × GCF(N, 16) = N × 16

Given that LCM(N, 16) = 48 and GCF(N, 16) = 8, we can substitute these values into the equation:

48 × 8 = N × 16

Solving for N:

N = × 48 × 8 / 16

N = × 24

Hence, the number N is 24.

The speed of the slower car is 84 km/h. This was calculated by using the distance equals rate times time formula, setting up an equation based on the combined distance both cars travel and the time they take to meet, and solving for the unknown rate.

Two cars leave towns 360 kilometers apart and travel toward each other; one car travels at a rate 12 kilometers per hour slower than the other. They meet in 2 hours, so we need to find the rate of the slower car. To solve this, we'll use the formula for distance which is rate × time. Let's denote the rate of the faster car as r and the rate of the slower car as r - 12. Since they meet in 2 hours, the faster car would have traveled 2r kilometers and the slower 2(r - 12) kilometers. The total distance covered by both cars should add up to 360 km, which gives us the equation 2r + 2(r - 12) = 360.

Simplifying the equation gives 4r - 24 = 360, and adding 24 to both sides gives 4r = 384. Dividing both sides by 4, we get r = 96. Therefore, the speed of the slower car, which is 12 km/h less than the faster car, is 96 - 12 = 84 km/h.

Answer:

c  ≤  c  ≤  [tex]\frac{-14}{17}[/tex].

Step-by-step explanation:

Given : −32c + 12 ≤ −66c − 16.

To find : Solve

Solution ": We have given

−32c + 12 ≤ −66c − 16.

On subtracting both sides by 12

- 32 c  ≤ −66c − 16 - 12

- 32 c  ≤ −66c − 28

On adding both sides by 66 c

-32c +66c  ≤  − 28.

34 c ≤  − 28.

On dividing both sides by 34

c  ≤  [tex]\frac{-28}{34}[/tex].

On dividing both number by 2

c≤  [tex]\frac{-14}{17}[/tex].

Therefore, c  ≤  [tex]\frac{-14}{17}[/tex].

The equation of the parabola with the vertex at (2,7) and passing through (-1,3) is y = -(4/9)(x - 2)^2 + 7, found by substituting the given points into the vertex form of a parabola's equation.

To find the equation of a parabola given its vertex and a point it passes through, we use the vertex form of a parabola's equation, which is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

Given the vertex at (2,7) and a point (-1,3) through which the parabola passes, we substitute these values into the vertex form to find the value of 'a'.

Substituting the vertex, we have:

y = a(x - 2)^2 + 7

Then, substituting the point (-1,3) into the equation, we get:

3 = a(-1 - 2)^2 + 7

Solving for 'a', we get:

3 = a(3)^2 + 7 \n3 = 9a + 7 \n-4 = 9a \na = -4/9

Therefore, the equation of the parabola is:

y = -(4/9)(x - 2)^2 + 7

The equation of the parabola with the vertex at (2,7) and passing through (-1,3) is y = -(4/9)(x - 2)2 + 7, found by substituting the given points into the vertex form of a parabola's equation.

To find the equation of a parabola given its vertex and a point it passes through, we use the vertex form of a parabola's equation, which is y = a(x - h)2 + k, where (h, k) is the vertex of the parabola.

Given the vertex at (2,7) and a point (-1,3) through which the parabola passes, we substitute these values into the vertex form to find the value of 'a'.

Substituting the vertex, we have:

y = a(x - 2)2 + 7Then, substituting the point (-1,3) into the equation, we get:

3 = a(-1 - 2)2 + 7

Solving for 'a', we get:

3 = a(3)2 + 7n3 = 9a + 7n-4 = 9ana = -4/9

Therefore, the equation of the parabolais: y=-(4/9)(x-2)2+7

Answer:

  You can use either of the following to find "a":

Step-by-step explanation:

It looks like you have an isosceles trapezoid with one base 12.6 ft and a height of 15 ft.

I find it reasonably convenient to find the length of x using the sine of the 70° angle:

  x = (15 ft)/sin(70°)

  x ≈ 15.96 ft

That is not what you asked, but this value is sufficiently different from what is marked on your diagram, that I thought it might be helpful.

__

Consider the diagram below. The relation between DE and AE can be written as ...

  DE/AE = tan(70°)

  AE = DE/tan(70°) = DE·tan(20°)

  AE = 15·tan(20°) ≈ 5.459554

Then the length EC is ...

  EC = AC - AE

  EC = 6.3 - DE·tan(20°) ≈ 0.840446

Now, we can find DC using the Pythagorean theorem:

  DC² = DE² + EC²

  DC = √(15² +0.840446²) ≈ 15.023527

  a ≈ 15.02 ft

_____

You can also make use of the Law of Cosines and the lengths x=AD and AC to find "a". (Do not round intermediate values from calculations.)

  DC² = AD² + AC² - 2·AD·AC·cos(A)

  a² = x² +6.3² -2·6.3x·cos(70°) ≈ 225.70635

  a = √225.70635 ≈ 15.0235 . . . feet

AB is bisected by CD (TRUE). This is True because E is the midpoint between A and B and CD passes through E

CD is bisected by AB (FALSE) CD is bisected by point F and not AB

AE = 1/2 * AB (TRUE) since E is the midpoint of AB , E divides AB into two equal halves

EF = 1/2 * ED (FALSE) The true statement would have been CF = 1/2* CD

FD = EB (FALSE) sinc we do not know if CD and AB are of the same lengths

CE + EF = ED (TRUE) since F is the midpoint the sum of CE and EF is equal to ED

The statements for the line AB and CD for this condition that are true are given as:

Option A: [tex]\overline{AB}[/tex] is bisected by [tex]\overline{CD}[/tex]

Option C: [tex]AE = \dfrac{1}{2} \times AB[/tex]

Option F: CE + EF = FD

A bisector of a line bisects that considered line. Bisect means to split in two equal parts.

For this case, we see that CD passes through mid point of AB, so CD is bisector of line AB or we say that line segment AB is bisected by line segment CD.

But AB does not passes through the center of AB, thus, AB is not a bisector of CD, or we say that line segment CD is not bisected by line segment AB

AE = EB

And AE + EB = AB

Thus, AE + AE + AB

or 2AE = AB

or AE = (AB)/2 = (1/2)AB

E is not necessary to be fixed on CD, it can move between C and F. Thus any statement about length of E to any point on CD is not necessary to be true.

FD is half of CD and EB is half of AB. It is not necessary that AB and CD are of same length, thus, it is not necessary that FD and EB are going to be of same length, thus, not congruent(two line segments are called congruent (denoted by ≅) if they are of same lengths).

CE + EF = CF, and CF = FD since F is midpoint.

Thus, CE + EF = FD

Thus, the statements for the line AB and CD for this condition that are true are given as:

Option A: [tex]\overline{AB}[/tex] is bisected by [tex]\overline{CD}[/tex]

Option C: [tex]AE = \dfrac{1}{2} \times AB[/tex]

Option F: CE + EF = FD

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Answer:

The correct option is D.

Step-by-step explanation:

From the given figure it is clear that two dots above 2, three dots above 3 five dots above 4, three dots above 5, and two dots above 6. It means the data set is

2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6

Total number of observations = 15

Therefore option D is correct.

15 is an odd number, so the median of the data is

[tex]Median=\frac{(\frac{n+1}{2})th}{2}[/tex]

[tex]Median=\frac{(\frac{15+1}{2})th}{2}=8th[/tex]

The 8th term of the data is 4, therefore the median of the data is 4. Option A is incorrect.

The mean of the data is

[tex]Mean=\frac{\sum x}{n}=\frac{2+2+3+3+3+4+4+4+4+4+5+5+5+6+6}{15}=\frac{60}{15}=4[/tex]

The mean of the data is 4. Option B is incorrect.

Range of the data is

[tex]Range=Maximum-Minimum[/tex]

[tex]Range=6-2=4[/tex]

Range of the data is 4. Option C is incorrect.