Karina read a total of 20 2/4 pages in her science and social studies books combined she read 12 3/4 pages in her science book how many pages did she read in her social studies book?

Answer:

The inverse of the function is [tex]y^{-1}=\sqrt{9-x}[/tex].

The domain of the inverse function is [tex]D:(-\infty,0],\{x|x\in \mathbb{R}\}[/tex]

Step-by-step explanation:

Given : Function [tex]y=9-x^2[/tex] where, [tex]x\geq 3[/tex]

To find : What is the inverse of the function? What is the domain of the inverse?

Solution :

Function [tex]y=9-x^2[/tex]

To find the inverse we interchange the value of x and y,

[tex]x=9-y^2[/tex]

Now, we get the value of y

[tex]y^2=9-x[/tex]

[tex]y=\pm\sqrt{9-x}[/tex]

As [tex]x\geq 3[/tex] so x>0

[tex]y=\sqrt{9-x}[/tex]

The inverse of the function is [tex]y^{-1}=\sqrt{9-x}[/tex].

The domain of the inverse is the range of the original function.

The range is defined as the set of all possible value of y.

As [tex]x\geq 3[/tex]

Squaring both side,

[tex]x^2\geq 9[/tex]

Subtract [tex]x^2[/tex] both side,

[tex]9-x^2\leq 0[/tex]

[tex]y\leq 0[/tex]

The range of the function is [tex]R:(-\infty,0],\{y|y\in \mathbb{R}\}[/tex]

The domain of the inverse function is [tex]D:(-\infty,0],\{x|x\in \mathbb{R}\}[/tex]

To determine whether the lines are parallel, coincident, skew, or intersecting, we first analyze the direction vectors of both lines obtained from the parametric equations. The lines are not parallel or coincident as they're not proportional. They could be intersecting or skew, which we can confirm by finding a potential point of intersection.

To determine the relationship between lines ℓ1 and ℓ2, we should analyze the direction vectors of both lines. The coefficients of t or u in the parametric equations for the lines refer to the direction vectors. For ℓ1, the direction vector is (-6, 9, -3). For ℓ2, it's (2, -3, 1).

Lines are parallel if their direction vectors are proportional, and intersect if they pass through a common point and their direction vectors are not proportional. Lines are coincedent if all corresponding points are identical, while skew lines are nonintersecting, nonparallel lines.

Given the direction vectors, we can see they are not proportional. Hence, ℓ1 and ℓ2 are neither parallel nor coincident. They might be intersecting or skew. To distinguish between these two, we need to try to find a point of intersection by equating ℓ1 and ℓ2 and solving for t and u. If a solution exists, the lines intersect at that specific point; if no solution exists, the lines are skew.

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Final answer:

Using the Law of Cosines, we find that the cosine of angle Z is approximately 0.6333, and therefore, angle Z in triangle XYZ is approximately 51° when rounded to the nearest degree.

Explanation:

To find the measure of angle Z in triangle XYZ, with sides XY = 15, YZ = 21, and XZ = 27, we can use the Law of Cosines. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. In our case, we can find the cosine of angle Z by the equation:

cos(Z) = (XY² + XZ² - YZ²) / (2 · XY · XZ)

Plugging in the values, we get:

cos(Z) = (15² + 27² - 21²) / (2 · 15 · 27)

Calculating further:

cos(Z) = (225 + 729 - 441) / (810)

cos(Z) = 513 / 810

cos(Z) = 0.6333

The next step is to find the angle whose cosine is 0.6333. We can use a calculator to find the inverse cosine:

Z = cos⁻¹(0.6333)

Hence, the measure of angle Z to the nearest degree is approximately:

Z ≈ 51°

The measure of angle Z is approximately 70 degrees (rounded to the nearest degree).

To find the measure of angle Z in triangle XYZ, you can use the Law of Cosines. The formula is:

[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]

where:

- c is the side opposite the angle you want to find (in this case, side XZ),

- a and b are the other two sides (XY and YZ),

- C is the angle opposite side c.

In this case, let C be the angle Z, and a = XY = 15, b = YZ = 21, and c = XZ = 27.

[tex]\[ 27^2 = 15^2 + 21^2 - 2(15)(21) \cos(Z) \][/tex]

Now, solve for cos(Z):

[tex]\[ 729 = 225 + 441 - 630 \cos(Z) \][/tex]

[tex]\[ 630 \cos(Z) = 441 - 225 \][/tex]

[tex]\[ 630 \cos(Z) = 216 \][/tex]

[tex]\[ \cos(Z) = \frac{216}{630} \][/tex]

[tex]\[ \cos(Z) \approx 0.343 \][/tex]

Now, find the angle Z:

[tex]\[ Z = \cos^{-1}(0.343) \][/tex]

[tex]\[ Z \approx 70.18 \][/tex]

So, the measure of angle Z is approximately 70 degrees (rounded to the nearest degree).

Answer:

9 DOGS

Step-by-step explanation:

5 LESS= 14-5

=9

The grain distributor can process 127.75 tons of grain in 8.75 hours.

To find out how much the distributor can process in 8.75 hours, you can multiply the amount of grain it can process in one hour by the number of hours. The distributor can process 14.6 tons of grain in an hour, so to find out how much it can process in 8.75 hours, you would multiply 14.6 by 8.75.

14.6 x 8.75 = 127.75

Therefore, the distributor can process 127.75 tons of grain in 8.75 hours.

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Answer:

[tex](x+12)^2=177[/tex]

Step-by-step explanation:

We have been given an equation: [tex]x^2+24x=33[/tex].

To complete the square we change the left hand side of the equation to a perfect square trinomial. For this we add [tex](\frac{b}{2})^2[/tex] to both sides of equation, where b is the coefficient of x.

We can see that coefficient of x for our given equation is 24. So we will add [tex](\frac{24}{2})^2=12^2=144[/tex] to both sides of our equation.

[tex]x^2+24x+144=33+144[/tex]

[tex]x^2+24x+144=177[/tex]

Let us factor left side of our equation as the square of  binomial.

[tex](x+12)^2=177[/tex]

Therefore, our resulting equation will be [tex](x+12)^2=177[/tex].

Answer: (x+12)^2=177

Step-by-step explanation:

Answer:

im pretty sure its 81

Step-by-step explanation:

The expression that have the following quotient is:

1)

[tex]\dfrac{-4x^2+20x-25}{-2x+5}[/tex]

It could also be written as:

[tex]\dfrac{-(5-2x)^2}{5-2x}\\\\\\=-(5-2x)\\\\\\=2x-5[/tex]

Hence, we get the quotient:

    [tex]2x-5[/tex]

Option: (1) is correct.

2)

[tex]\dfrac{-14x^2+35x}{-7x}[/tex]

which is simplified as follows:

[tex]\dfrac{-7x(2x-5)}{-7x}\\\\\\=2x-5[/tex]

Option: (2) is correct.

3)

[tex]\dfrac{12x^2-58x+70}{6x-14}[/tex]

which is simplified as follows:

[tex]=\dfrac{2(2x-5)(3x-7)}{2(3x-7)}\\\\\\=2x-5[/tex]

Hence, option: (3) is correct.

4)

[tex]\dfrac{6x^2-10x-4}{3x+1}[/tex]

On simplifying:

[tex]\dfrac{2(3x+1)(x-2)}{3x+1}\\\\\\=2(x-2)\\\\\\=2x-4[/tex]

Hence, option: (4) is incorrect.

5)

[tex]\dfrac{18x-45}{9x}[/tex]

on simplifying:

[tex]\dfrac{9(2x-5)}{9x}\\\\\\=\dfrac{2x-5}{x}[/tex]

Hence, option: (4) is incorrect.

Final answer:

To find the general solution of the given system of differential equations, first, solve the second equation for x1 in terms of x2. Then substitute x1 into the first equation and simplify to isolate x'2. Next, solve the second equation for x2 in terms of x1 and substitute x2 into the first equation. Simplify and rearrange to isolate x'1. Finally, solve the resulting differential equations to find the general solution.

Explanation:

To find the general solution of the given system of differential equations:
x'1 = 3x1 - x2 + et, x'2 = x1

Step 1: Solve the second equation for x1 in terms of x2:
x1 = x'2

Step 2: Substitute x1 into the first equation:
x'1 = 3(x'2) - x2 + et

Step 3: Simplify and rearrange the equation to isolate x'2:
x'2 = (x'1 + x2 - et)/3

Step 4: Solve the second equation for x2 in terms of x1:
x2 = x'1 - 3x'2 + et

Step 5: Substitute x2 into the first equation:
x'1 = 3x1 - (x'1 - 3x'2 + et) + et

Step 6: Simplify and rearrange the equation to isolate x'1:
x'1 = (3x'2 + x'1 - 2et)/3

Now we have two differential equations for x'1 and x'2 in terms of each other. These can be solved using standard methods to find the general solution.

The probability is 1/12.

Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e. how likely they are to happen, using it.

Given:

Total children = 12

There is possibility of girl number 11.

So, the probability that child number 11 is female

=1/12.

Hence, the probability is 1/12.

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Final answer:

The probability that the 11th child is female is 50%, assuming an equal chance for the birth of either sex and that the gender of the previous children does not affect this outcome.

Explanation:

The question at hand is a basic probability question in the realm of mathematics. It attempts to find the likelihood of an event that is purely random and independent of previous outcomes. The event in question is the sex of a newborn child, which can either be male or female, generally with equal likelihood.

To determine the probability that child number 11 is female, we assume a 50% chance for either sex, since the likelihood of having a boy or girl is typically equal. The sex of the previous children has no influence on the gender of the next child. Therefore, the probability of child number 11 being female is simply 1/2 or 50%.

It is important to understand that each child's sex is determined independently of their siblings. This is akin to flipping a fair coin where each flip is independent of the previous flips.

Answer:

aₙ = a₁ + (n-1).d

103

107

Step-by-step explanation:

This is an arithmetic sequence because the difference (d) between 2 successive numbers is constant.

d = 91 - 87 = 95 - 91 = 99 - 95 = 4

If the first term is a₁, we can find the n term using the following expression.

aₙ = a₁ + (n-1).d

Here, we want to find the fifth and sixth elements.

a₅ = 87 + (5-1).4 = 103

a₆ = 87 + (6-1).4 = 107

To expand and evaluate the summation notation with an absolute value, we can use the binomial theorem. The expanded form of the series will depend on whether the sum inside the absolute value is positive or negative. Substituting the given values and simplifying the expression will give the sum of the series.

A binomial expansion is a way of expressing an algebraic quantity as a sum of an infinite series of terms. In this case, we have an absolute value in the summation expression.

To expand and evaluate the summation notation, we can use the binomial theorem.

The binomial theorem states that (a + b)^n = C(n,0)a^n + C(n,1)a^(n-1)b + C(n,2)a^(n-2)b^2 + ... + C(n,n)b^n, where C(n,k) represents the binomial coefficient.

In the given case, we have an expression of the form |a + b|.

To expand this, we can consider two cases: a + b ≥ 0 and a + b < 0. If a + b ≥ 0, then |a + b| = a + b. If a + b < 0, then |a + b| = -(a + b).

Thus, the expanded form of the series will be a + b when a + b ≥ 0, and -(a + b) when a + b < 0.

To evaluate the series, substitute the given values for a and b into the expanded form and simplify the expression.

Answer:

15 | 10-dollar shares, 5 | 12- dollar shares

Step-by-step explanation:

Write the equation by adding the total values of each type of share.

10(2t+5)+12t=210

We can solve this equation for t to find the number of $12 shares.

10(2t+5)+12t=210

20t+50+12t=210

32t=160

t=5

So, the number of $12 shares is 5. Since the number of $10 shares is 5 more than twice the number of $12 shares, the number of $10 shares is 2(5)+5=15.

Hilda bought 5 $12 shares and 15 $10 shares.

Final answer:

To find the length of each leg of an isosceles triangle with an area of 125 ft² and a base of 16 ft, we first calculate the height and then apply the Pythagorean Theorem. The length of each leg is approximately 17.6 ft.

Explanation:

The question asks about finding the length of each leg of an isosceles triangle given the area and the base. To solve this, we use the formula for the area of a triangle, A = 1/2 × base × height, and apply the Pythagorean Theorem since the height creates a right triangle with the leg of the isosceles triangle and half of the base.

First, we find the height using the given area, A = 125 ft² and base, b = 16 ft. The formula rearranges to height = 2A / base, giving us height = 2(125) / 16 = 15.625 ft.

Next, using the Pythagorean Theorem, a² + b² = c², we let a represent half of the base (8 ft), b be the height (15.625 ft), and c be the leg of the triangle. Solving for c, we find c ≈ 17.6 ft, rounded to the nearest tenth.

Answer:

[tex]A= \frac{400}{21} B=\frac{580}{21}[/tex]

Step-by-step explanation:

You just have to remember the rules of the triangles in order to be able to solve this:

The rule of trigonometry that you will have to use is a is similie to 20, B is simile to 29 and 20 is 20.

So you would have to put them like this:

[tex]\frac{21}{20}= \frac{20}{a}= \frac{29}{b}[/tex]

You just have to clear the equations for A and B:

[tex]a=\frac{20*20}{21}=\frac{400}{21}\\  b=\frac{29*20}{21} =\frac{580}{21}[/tex]

So that would be your answer.

The value of 'a' is 400/21 and the value of 'b' is 580/21 and this can be determined by using the properties of trigonometry.

According to the properties of trigonometry:

[tex]\dfrac{21}{20}=\dfrac{20}{a}=\dfrac{29}{b}[/tex]    --- (1)

Now, in order to determine the value of 'a' use the above equation:

[tex]\dfrac{21}{20}=\dfrac{20}{a}[/tex]

Cross multiply in the above equation.

[tex]21a = 20\times 20[/tex]

21a = 400

Divide 400 by 21 in order to get the value of a.

[tex]a = \dfrac{400}{21}[/tex]

Now, to determine the value of 'b' again use the equation (1).

[tex]\dfrac{20}{\dfrac{400}{21}}=\dfrac{29}{b}[/tex]

Cross multiply in the above expression.

[tex]20b = 29\times \dfrac{400}{21}[/tex]

[tex]20b = \dfrac{11600}{21}[/tex]

Now, divide on both sides by 20 in the above expression.

[tex]\rm b = \dfrac{11600}{21\times 20}[/tex]

[tex]\rm b = \dfrac{580}{21}[/tex]

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The function ff(x) = 0.75(1.05)ˣ is an example of an exponential growth function.

The percent rate of change is 5%

What type of function is the equation

From the question, we have the following parameters that can be used in our computation:

f(x) = 0.75(1.05)ˣ

The function ff(x) = 0.75(1.05)ˣ is an example of an exponential growth function.

In exponential growth functions, the base (in this case, 1.05) is greater than 1, and as x increases, the function increases

Also, we have the percent rate of change to be

Rate = 1.05 - 1

Rate = 0.05

So, we have

Rate = 5%

Hence, the percent rate of change is 5%