CORE
SKILL
ADVANCED
If a is inversely related to the square of b, and a = 3 when bis
8, which of the following is another possible value for a and b?
(A) a = 4, b = 12
(B) a = 12, b = 4
(C) a = 8, b = 3
(D) a = 2, b = 4
(E) a = 6, b = 4

The area of the largest rectangle is 25 square units.

It is given that the radius of the semicircle is [tex]4[/tex].

It is required to inscribe a rectangle of maximum area.

Consider a rectangle of sides [tex]x[/tex], [tex]y[/tex] as shown in the given figure.

From the attached figure,[tex]AB=x[/tex], [tex]OB=r[/tex] and [tex]OB=\frac{y}{2}[/tex].

Now, in the triangle OAB, apply Pythagoras' theorem as,

[tex](OA)^{2}=(OB)^{2}+(AB)^2[/tex]

[tex]r^2=(\frac{y}{2} )^2+x^2\\\\[/tex]

[tex]16=\frac{y}{4}+x^2[/tex]

[tex]y^2=4(16-x^2)[/tex]

[tex]y=2\sqrt{16-x^2}[/tex]

So, the area of the rectangle will be,

Area,

[tex]A = 2x \times y[/tex]

= [tex]2x \times \sqrt{(16 - x^2)}[/tex]

Differentiate the area with respect to ,

[tex]A' = 2 \times \sqrt{(16 - x^2)} - 2x^2/(16 - x^2)[/tex]

Differentiate the area twice as,

When[tex]x = 0, y = 5[/tex] and when [tex]x = 5, y = 0[/tex], [tex]area = 0[/tex].

It implies that area is maximum when the value of x lies between 0 and 5.

This will occur where [tex]A'= 0[/tex].

⇒ [tex]2 \times \sqrt{(16 - x2) }- 2x^2/(16 - x^2) = 0[/tex]

⇒[tex]2 \times \sqrt{(16 - x^2)} = 2x^2/(16 - x^2)[/tex]

On simplification, we get,

⇒ [tex]2 \times (16 - x^2) = 2x^2[/tex]

⇒ [tex]16 - x^2 = x^2[/tex]

⇒ [tex]2x^2= 16[/tex]

⇒ [tex]x^2 = 16/2[/tex]

⇒ [tex]x = \sqrt{8}[/tex]

Now, [tex]y = \sqrt{(16 - x^2)}[/tex] becomes

⇒[tex]y = \sqrt{(16 - (\sqrt8)^2)}[/tex]

⇒ [tex]y = \sqrt{(16 - 8)}[/tex]

[tex]y =\sqrt{8}[/tex]

Maximum area = 2xy

[tex]=2(\sqrt{8)}(\sqrt{8})[/tex]

[tex]=16[/tex]

Therefore, the area of the largest rectangle is 16 square units.

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The length of AC is 17.3 cm

acc to our question,

Hence,The length of AC is 17.3 cm

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Answer:

  no

Step-by-step explanation:

You want to know if side lengths 26 in, 36 in, 46 in form a right triangle.

The integer side lengths will form a right triangle if they are a Pythagorean triple, or a multiple of one.

We notice that these side lengths have a common difference of 10 in. That makes them an arithmetic sequence. The only Pythagorean triples that are an arithmetic sequence are {3, 4, 5} and its multiples. The shortest side is 3 times the common difference.

The lengths {26, 36, 46} cannot form a right triangle. The attachment shows the triangle is an obtuse triangle.

No, the triangle is not a right triangle.

__

Additional comment

The side lengths will form a right triangle if and only if they satisfy the relation a² +b² = c², where a, b are the two short sides.

  26² +36² = 1972 ≠ 2116 = 46²

The area of the circle on the paper is 4.5 cm when traced around the edge of the quarter on a sheet of paper.

The area of the circle is equal to the product of the square of the radius of the circle and pi.

A = πr²

where 'r' is the radius of the circle

The diameter of a quarter is 2.4 cm. You trace around the edge of the quarter on a sheet of paper.

The area of the circle on the paper = πr²

Here r = (diameter of a quarter)/2 = 2.4/2 = 1.2 cm

The area of the circle on the paper = π(1.2)²

The area of the circle on the paper = π × 1.44

The area of the circle on the paper = 4.5238

Round the answer to the nearest tenth.

The area of the circle on the paper = 4.5 cm

Thus, the area of the circle on the paper is 4.5 cm

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Answer:its 4.5

Step-by-step explanation:

Answer:

[tex]9b + 10u > 1400[/tex]

The a < 6 or a>-9 compound inequality is different than the rest as it involves one interval instead of a reunion of two intervals.

A phrase having two inequality statements connected by the words "or" or "and" is referred to as a compound inequality. The conjunction "and" denotes the simultaneous truth of both truths in the compound phrase. It is when the solution sets for the various assertions to cross over or intersect.

When the word and is used to connect two inequalities, the compound inequality is solved when both inequalities are true at the same time. It is the intersection or overlap of each inequality's solutions.

1)(−∞,−3)∪(4,∞)

2)(−∞,−2)∪(8,∞)

3)(−∞,−5)∪(7,∞)

4)(−∞,6)∪(−9,∞)=(−9,6)

The fourth compound inequality is different than the rest as it involves one interval instead of a reunion of two intervals.

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The factor of the expression 72[tex]a^{9} b^{3}[/tex] - 18[tex]a^{7} b^{5}[/tex] is 18[tex]a^{7} b^{3}[/tex](4[tex]a^{2}[/tex] - [tex]b^{2}[/tex]).

Given expression:

= 72[tex]a^{9} b^{3}[/tex] - 18[tex]a^{7} b^{5}[/tex]

divide by 18 .

= 4[tex]a^{9} b^{3}[/tex] - [tex]a^{7} b^{5}[/tex]

divide by [tex]a^{7}[/tex]

= 4[tex]a^{2}b^{3}[/tex] - [tex]b^{5}[/tex]

divide by [tex]b^{3}[/tex]

= 4[tex]a^{2}[/tex] - [tex]b^{2}[/tex]

= 18[tex]a^{7} b^{3}[/tex](4[tex]a^{2}[/tex] - [tex]b^{2}[/tex]).

Therefore the factor of the expression 72[tex]a^{9} b^{3}[/tex] - 18[tex]a^{7} b^{5}[/tex] is 18[tex]a^{7} b^{3}[/tex](4[tex]a^{2}[/tex] - [tex]b^{2}[/tex]).

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Using simple mathematical operations, the missing values are as follows:

(B) Dijon Mustard: 18g

(C) Barbeque sauce: 70g

(D) Worcestershire sauce: 10.35g

(E) Kosher salt: 1.5g

(F) Ground black pepper: 2.3g

(G) Crushed red pepper: 1.15g

So, we can calculate missing values as follows:

(B) Dijon Mustard: 18g

(C) Barbeque sauce: 70g

(D) Worcestershire sauce: 10.35g

(E) Kosher salt: 1.5g

(F) Ground black pepper: 2.3g

(G) Crushed red pepper: 1.15g

Therefore, using simple mathematical operations, the missing values are as follows:

(B) Dijon Mustard: 18g

(C) Barbeque sauce: 70g

(D) Worcestershire sauce: 10.35g

(E) Kosher salt: 1.5g

(F) Ground black pepper: 2.3g

(G) Crushed red pepper: 1.15g

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Answer:

32 books

Step-by-step explanation:

Well if each have 8 classes and 2 books per class then to get the total of books you'd multiply 8x2 and you get 16 for one alone. Since its two people sharing a single locker than add 16 with another 16 and you get 32!

sorry I can't physically show the work but writing down a 8x2 on the side and then adding 16 to that answer should show it off! :]

Using percentages we calculate in the beginning there were 125 jellybeans in the jar.

Let the number of jellybeans in the jar was x.

Now student ate 20% of the jellybeans.

Number of jellybeans students ate = 20% of x = 0.2x

Number of jellybeans left = x - 0.2x = 0.8x

But the number of jellybeans is given as 100.

Therefore 0.8x = 100

or, x = 100 /0.8 = 125

Hence there were 125 jellybeans in the jar.

The English word "percentage" derives from the Latin phrase "per centum," which means "by the hundred." Percentages are fractions with a denominator of 100.

In other words, that is a relationship where it is always assumed that the value of the whole is 100.

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Answer: a) = 4

b) = 30

Step-by-step explanation:

a) They are asking for the range of the marks so you do 10-6 as 10 is the highest mark and 6 is the lowest

b) You find this out by adding the frequency up 10+4+7+4+5 = 30

Answer: z=-2

Step-by-step explanation:

3z+6z-30=-48

9z-30=-48

9z-30=-48+30

9z=-18

z=-2

Answer:

The tablet costs more than the smartphone

Step-by-step explanation:

1 year tablet = $1.50

1 year smartphone = $0.02x12months = $0.24

[tex]\fbox{90}[/tex]

In order to find 36 is 40% of what number, we will need to use the following formula:

[tex]number\times100/Percent[/tex]

[tex]\frac{36\times100}{40}[/tex]

[tex]36\times100=3,600[/tex]

[tex]3,600\div40[/tex]

[tex]=90[/tex]

The solution of the number in the expression ''40% of a number is 36'' is 90.

Given that,

To find the solution for 40% of that number is 36.

Used the concpet of percentage which stated that,

A number or ratio that can be expressed as a fraction of 100 or a relative value indicating the hundredth part of any quantity is called a percentage.

Let us assume that the solution of the given condition is x.

So, it can be written as,

[tex]40\% \times x = 36[/tex]

[tex]\dfrac{40}{100} \times x = 36[/tex]

Multiply both sides by 100,

[tex]40x = 36 \times 100[/tex]

[tex]40x = 3600[/tex]

Divide both sides by 40,

[tex]x = \dfrac{3600}{40}[/tex]

[tex]x = 90[/tex]

So, the solution for the number is 90.

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Answer:

Step 6 is wrong

Answer:  The correct answer is Option D: h=2A/(b1+b2)

Step-by-step explanation:

Solve for h:

a=(1/2h)(b1+b2)

Step 1: Flip the equation

1/2b^2h+1/2b1h=a

Step 2: Factor out variable h

h((b2+b1)/2)=a

Step 3: Divide both sides by (b^2+b1)/2

h((b2+b1/2))/(b2+b1/2)=(a/(b2+b1)/2)

h=2a/(b2+b1)  

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Answer:

see explanation

Step-by-step explanation:

- 1, 2, 5, 8

(a)

the common difference

d = a₂ - a₁ = 2 - (- 1) = 2 + 1 = 3

(b)

to find a term in the sequence add d = 3 to the previous term

a₅ = = 8 + 3 = 11

a₆ = 11 + 3 = 14

a₇ = 14 + 3 = 17

a₈ = 17 + 3 = 20

(c)

the recursive rule is

[tex]a_{n}[/tex] = [tex]a_{n-1}[/tex] + 3 : a₁ = - 1

(d)

the explicit rule for an arithmetic sequence is

[tex]a_{n}[/tex] = a₁ + (n - 1)d

where a₁ is the first term and d the common difference

here a₁ = - 1 and d = 3 , then

[tex]a_{n}[/tex] = - 1 + 3(n - 1) = - 1 + 3n - 3 = 3n - 4

Answer:

V<4400

Step-by-step explanation:

Answer:

the answer is V<4,400

if you are using khan academy, do not add the comma for 4,400, otherwise it will not work.

khan academy answer: V<4400

Step-by-step explanation:

if the air leaked out, which is V, that means V would be less than 4,400 because 4,400 is what we have started with.

Therefore, the inequality would be V<4,400 or for khan academy, V=4400

Answer:

85 ggoooofffffyyy

Step-by-step explanation:

7. The composite functions are given as follows:

a. f(g(x)) = [tex]\frac{3x}{-x^2 + 3x - 1}[/tex].

b. g(f(x)) = [tex]\frac{3x(x - 1)}{9x^2 + x - 1}[/tex]

c. (f(g(3)) = -9.

8. The inverse function is given as follows:

[tex]f^{-1}(x) = \frac{4 + x}{3 - x}[/tex]

The composite functions are obtained using the inner function as the input to the outer function.

In this problem, the functions are defined as follows:

The composite of f and g is:

[tex]f(g(x)) = f\left(\frac{3x}{x^2 + 1}\right) = \frac{\frac{3x}{x^2+1}}{\frac{3x}{x^2+1} - 1} = \frac{\frac{3x}{x^2+1}}{\frac{-x^2 + 3x - 1}{x^2 + 1}} = \frac{3x}{-x^2 + 3x - 1}[/tex]

At x = 3, the numeric value of the composite function is given as follows:

f(g(3)) = 9/(-9 + 9 - 1) = -9.

The composite function of g and f is given as follows:

[tex]g(f(x)) = g\left(\frac{x}{x - 1}\right) = \frac{\frac{3x}{x - 1}}{\left(\frac{3x}{x - 1}\right)^2 + 1} = \frac{3x}{x - 1} \times \frac{(x - 1)^2}{9x^2 + x - 1} = \frac{3x(x - 1)}{9x^2 + x - 1}[/tex]

The function is given as follows:

y = (3x + 4)/(x - 1)

The find the inverse, the variables x and y are exchanged, and then y is isolated, as follows:

x = (3y + 4)(y - 1)

xy - x = 3y + 4

3y - xy = 4 + x

y(3 - x) = 4 + x

y = (4 + x)/(3 - x)

[tex]f^{-1}(x) = \frac{4 + x}{3 - x}[/tex]

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Answer:

Step-by-step explanation:

16/(2+6)

16/8=2

Answer:

see explanation

Step-by-step explanation:

there is a common difference between consecutive terms in the sequence, that is

8 - 13 = 3 - 8 = - 2 - 3 = - 7 - (- 2) = - 5

this indicates the sequence is arithmetic with nth term

[tex]a_{n}[/tex] = a₁ + (n - 1)d

where a₁ is the first term and d the common difference

here a₁ = 13 and d = - 5 , then

[tex]a_{n}[/tex] = 13 - 5(n - 1) = 13 - 5n + 5 = 18 - 5n

then

a₁₀₀ = 18 - 5(100) = 18 - 500 = - 482

The perimeter of the composite figure would be = 21 (to the nearest tenth).

Perimeter is defined as the quantity that is used to evaluate the total length of the sides of an object.

From the given diagram, the composite figure is made up of 6 sides. The addition of these sides is the figures perimeter.

The sides of the composite figure are:

a = 6; b = 12; c = 9; d = 9; e = -9; f = -6.

The perimeter = a + b + c + d + e + f.

= 6 + 12 + 9 +9+ (-9) + (-6)

= 27 - 6

= 21 ( to the nearest tenth)

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Answer:

5 and 4

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the gradient and c the y- intercept )

(a)

y = 5x - 3 ← is in slope- intercept form

with gradient m = 5

(b)

3y - 12x + 7 = 0 ( add 12x to both sides )

3y + 7 = 12x ( subtract 7 from both sides )

3y = 12x - 7 ( divide through by 3 )

y = 4x - [tex]\frac{7}{3}[/tex] ← in slope- intercept form

with gradient m = 4

In given word problem ending time of the play is 10.50 AM.

What is word problem?

 A comprehensive list of math word problems focuses on addition, subtraction, multiplication, division and more specific operations. life scenario. This means that children must be familiar with vocabulary associated with familiar mathematical symbols in order to understand word problems.

Here starting time of play = 9:05 AM

It took 1 hour 45 minutes to end the play. Then ,

=> 9 hour  05 minutes

    1 hour   45 minutes +

=> 10 hour 50 minutes.

Hence ending time of the play is 10.50 AM.

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Answer:

infinite solution

Step-by-step explanation:

Since 0 = 0 for any value of y, the system of equations has infinite solutions.

The equation of the line that passes through the points  (9,2) and (3,4) is x + 3y = 15  and the graph  is shown below .

In the question ,

it is given that

the required line passes through the points (9,2) and (3,4)

in order to find the equation of  line we first need to find slope .

so , the slope(m) will be

m = (4-2)/(3-9)

m = 2/(-6)

m = -1/3

The equation of the line passing through the point (3,4) having slope -1/3 is

y - 4 = (-1/3)(x - 3)

3(y - 4) = -1(x - 3)

3y - 12 = -x + 3

x + 3y = 12 + 3

x + 3y = 15

the graph of the line is drawn below .

Therefore ,The equation of the line that passes through the points  (9,2) and (3,4) is x + 3y = 15  .

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Step-by-step explanation:

9 × 1/2 = 4.5

8 × 1/2 = 4

hope it's helpful

The value of y that makes the two matrices inverse of each other is 1/2

The inverse of a square matrix A, denoted by A⁻¹.

The formula is [tex]\frac{1}{|A|}.adj(A)[/tex]

The product of A and A⁻¹  should be the identity matrix. The identity matrix that results will be the same size as matrix A.

Inverse of given first matrix is

[tex]\left[\begin{array}{cc}6&14\\-2&-4\end{array}\right][/tex]

Determinant = |A|  = 6(-4) - (14)(-2)

= -24 + 28

= 4

Adjoint of matrix = [tex]\left[\begin{array}{cc}-4&-14\\2&6\end{array}\right][/tex]

Inverse of matrix = [tex]\frac{1}{|A|}.adj(A)[/tex]

Therefore , Inverse = [tex]\frac{1}{4}.\left[\begin{array}{cc}-4&-14\\2&6\end{array}\right][/tex]

= [tex]\left[\begin{array}{cc}-4/4&-14/4\\2/4&6/4\end{array}\right][/tex]

A⁻¹ = [tex]\left[\begin{array}{cc}-1&-7/2\\1/2&3/2\end{array}\right][/tex]

Is required inverse

If we compare the inverse with second matrix

y = 1 / 2

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Answer:

third option:      1/2

Step-by-step explanation: