What are two solutions of 3 + 4 |x/2 + 3| = 11?

The required equation is  70 = (1/2) × base × 8.75, resulting in a base length of 16 inches.

To find the length of the base of a triangle when given the area and the height, we use the formula for the area of a triangle: Area = (1/2) × base × height. In this problem, the area is 70 square inches and the height is 8 3/4 inches.

First, convert the height from a mixed number to an improper fraction or decimal. We have:

8 3/4 inches = 8 + 3/4 = 8.75 inches

Next, write the equation based on the area formula:

70 = (1/2) × base × 8.75

To solve for the base, first multiply both sides of the equation by 2 to eliminate the fraction:

140 = base × 8.75

Then, divide both sides by 8.75 to isolate the base:

base = 140 / 8.75

Calculating this gives:

base = 16 inches

Therefore, the length of the base of the triangle is 16 inches.

Answer:Option C  : 56 °

Step-by-step explanation:

In the given circle the chords JH and KI  intersect outside the circle forming <JGK.

The Angle of intersecting chords Theorem states:

The measure of the angle formed by 2 chords that intersect inside the circle is 1÷2 the sum of the chords' intercepted arcs.

m<JGK=[tex]\frac{125-13}{2} =\frac{112}{2} =56[/tex]°

m<JGK=56°

For a better understanding of the explanation provided here, please go through the graphs in the attached files. The first file contains the explanations to the first part of the question and the second file contains the three graphs required to explain the second part (the y intercept part) of the question.

It has been given to us that the pH concentration equation is:

[tex] p(t)=-log_{10}t [/tex]

where, The variable t represents the amount of hydronium ions and p(t) gives the resulting pH level.

As can be seen from the first graph, pH is 1 when the amount of hydronium ions, t is 0.1. This is as expected, because, from the concept of logarithms we know that:

[tex] log_{10}(0.1)=log_{10}(\frac{1}{10})=log_{10}(10^{-1})=-1\times log_{10}10=-1\times 1=-1 [/tex]

Thus, [tex] p(0.1)=-1\times-1=1 [/tex]

Again, the value of pH, p(t) is 0 when the concentration of the Hydronium ions, t is 1. This, again, is as expected, because, from the concept of logarithms we know that [tex] log(1)=0 [/tex].

If the pool maintenance man uses our graph and needs to find the the pH level if the amount of hydronium ions is raised to 0.50, then to do so, the pool maintenance man must draw a vertical line from t=0.5 and check the x coordinate of the point where this vertical line intersects the p(t) graph. As we can see from the graph this roughly comes out to be the horizontal line y=0.3.

Let us now move on to the second part of the question. For a better understanding of this part please go through the second graph which has been attached. This graph contains all translations on the original graph and the explanation to each one of them is provided below:

Let us start with the last one, -p(t). We know that, since, [tex] p(t)=-log_{10}(t) [/tex], therefore, [tex] -p(t)=log_{10}(t) [/tex], simply a vertical "flip" or reflection of the original graph. And because the original graph did not have a y intercept, this graph too will not have a y intercept.

Let us now move on to the first one. Here, [tex] p(t)+1=-log(t)+1 [/tex]. This graph will simply move the original graph vertically up by 1 unit and that will not have any bearing on the y intercept and thus, the graph of p(t)+1 will again, not have a y intercept.

Finally, let us move on to the graph of the equation p(t+1). As can be seen, this graph has a y intercept at the origin when t=0. This is because:

[tex] p(t+1)=log_{10}(t+1) [/tex]

and when t=0, [tex] p(t+1)=p(0+1)=log_{10}(0+1) [/tex]

[tex] p(1)=log_{10}(1)=0 [/tex]

Thus, when t=0, p(t+1)=0 too and thus we have a y intercept.

Therefore, out of the three transformations, only the middle transformation results in a y-intercept, the explanation for which has been provided.

Final answer:

To find the three consecutive odd integers whose sum is 327, we set up an equation with n as the first odd integer, resulting in the integers being 107, 109, and 111.

Explanation:

The question asks for the three consecutive odd integers whose sum is equal to 327. To solve this, we can represent the first integer as n, the second as n+2, and the third as n+4 because odd numbers are two units apart. We can then set up the following equation to find the value of n:

n + (n + 2) + (n + 4) = 327

Simplifying this equation, we get:

3n + 6 = 327

Subtracting 6 from both sides gives us:

3n = 321

Dividing both sides by 3, we find that n = 107. Therefore, the three consecutive odd integers are 107, 109, and 111.

The square root of -12, an imaginary number, can be simplified to 2i√3. This involves breaking -12 into -1 * 12 and taking the square root of both parts to get 'i' and 2i√3 respectively.

The question asked is to simplify the imaginary number square root of -12. Imaginary numbers are those numbers whose square is less than or equal to zero. They are expressed in the form of 'i' where 'i' is the imaginary unit with the property i^2 = -1. When we take square root of a negative number, 'i' is involved in the solution.

To simplify the square root of -12, we can express -12 as -1 * 12. In this case, square root of -1 is 'i' and square root of 12 is 2 * square root of 3 (since 12= 4*3 and square root of 4 is 2).

Therefore, square root of -12 can be written as 'i' * 2 * square root of 3 or simply 2i√3.

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Answer:

Option 4th is correct.

Step-by-step explanation:

We have been the information 25 students play an instrument 20 are in a band 30 are not in a band.

We need to create a two way table:

                                      Band            Not in Band       Total

Play instrument                20                   5                     25

Do not play instrument     0                     25                 25

Total                                  20                    30                50

Option 4th is correct.

Solving the quadratic equation 2x² + 3x = - 8 for x, x = - 3/4 ± i√55/4

To solve 2x² + 3x = - 8, we proceed as follows.

Since we have the quadratic equation 2x² + 3x = - 8, to solve by completing the square, we rewrite the equation as

2x²/2 + 3x/2 = - 8/2

x² + 3x/2 = - 4

Next, we add the square of half the coefficient of x to both sides. So, we have that

x² + 3x/2 + (3/2 ÷ 2)² = - 4 + (3/2 ÷ 2)²

x² + 3x/2 + (3/2 × 1/2)² = - 4 + (3/2 × 1/2)²

x² + 3x/2 + (3/4)² = - 4 + (3/4)²

(x + 3/4)² = - 4 + 9/16

(x + 3/4)² = (-64 + 9)/16

Now, taking the square root of both sides, we have that

(x + 3/4)² = -55/16

√(x + 3/4)² = ±√(-55/16)

x + 3/4 = ±√-55/4

x + 3/4 = ±i√55/4

x = - 3/4 ± i√55/4

So, the value of x = - 3/4 ± i√55/4

Answer: 90° option a

180° option b

Step-by-step explanation: If we do a rotation of 90° about the X point, this means that the angle between the lines XZ and X'Z must be equal to 90°.

The options that have a 90° angle between these two lines are a and c.

And here you can see that, if the rotation is counterclockwise, the option a is a rotation of 90° and option c is a rotation of 270°, while if the rotation is defined clockwise, the option c is 90° and option a is 270°.

Usually, the rotations are defined as counterclockwise, so under this viewpoint, the right answer is option a.

For a 180° rotation the angle between XZ and X'Z' must be equal to 180°, the only option that has a 180° angle is option b, so here b is the answer.

To find the image of triangle XYZ after a 90 degree rotation followed by a 180 degree rotation about point X, we perform the rotations successively, leading to a total rotation of 270 degrees counterclockwise about point X.

To identify the image of the triangle XYZ after a composition of a 90 degree rotation and a 180 degree rotation, both about point X, we must perform the transformations step by step.

Firstly, we will apply a 90 degree rotation around point X. Imagine triangle XYZ in a coordinate plane where X is the origin. A 90 degree rotation would result in each point (x,y) of the triangle moving to (-y, x). The identity of triangle XYZ changes as each point follows this rule.

Next, we will apply a 180 degree rotation around point X. A 180 degree rotation is the same as an inversion of a point at (x,y) to (-x,-y). Again, each point of the already transformed triangle will follow this rule.

Overall, the composition of these two rotations is effectively a single rotation by 90 + 180 = 270 degrees. Triangle XYZ will thus have been rotated 270 degrees counterclockwise around point X to its final image position.

Final answer:

The total cost for n uniforms is $65.98 * n + $70.

We use the inequality 65.98n + 70 ≤ 550 to figure out how many uniforms the owner can afford.

Without the logo fee, the owner can afford 8 uniforms since the cost is then less than $550.

Explanation:

To assist with the question regarding the cost of uniforms, let's look at each part individually:

A. Cost of uniforms

The cost of a jacket is $37.99 and the pants are $27.99. The restaurant's logo is a one-time fee of $70.00. So for n uniforms, the cost would be:

Total cost for n uniforms = (Cost of jacket + Cost of pants) * n + Logo fee
= ($37.99 + $27.99) * n + $70.00
= $65.98 * n + $70.00.

B. Inequality for the number of uniforms

We need to ensure the total cost of uniforms is less than or equal to $550. Thus, the inequality is:
65.98n + 70 ≤ 550.

C. Buying 8 uniforms

To determine if the owner can buy 8 uniforms, we substitute n with 8 in the cost expression:
65.98 * 8 + $70 = $597.84 + $70 = $667.84, which is more than $550.

D. Without the logo fee

If the logo is not added, the cost for 8 uniforms would be just the sum of the jackets and pants:
$65.98 * 8 = $527.84, which is less than $550, so yes, without the logo fee, 8 uniforms can be afforded.

From the given graph the length of the AA' is 5.39 units.

A pair of numbers called coordinates are used to locate a point or a form in a two-dimensional plane. The x-coordinate and the y-coordinate are two numbers that define a point's location on a 2D plane.

To find the length of AA', we can use the distance formula:

[tex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex]

where (x₁, y₁) = (0, 0) is the coordinates of point A, and (x₂, y₂) = (5, 2) is the coordinates of point A'.

Substituting the values, we get:

[tex]d = \sqrt{(5 - 0)^2 + (2 - 0)^2}\\d = \sqrt{25 + 4}\\d = \sqrt{29}[/tex]

Therefore, the length of AA' is approximately 5.39 units (rounded to two decimal places).

Learn more about coordinates here:

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