Angles α and β are the two acute angles in a right triangle. Use the relationship between sine and cosine to find the value of β if β < α.


sin(7x − 15) = cos(3x + 5)

Information Technology and Engineering

Mathematics is also a branch of Science. A person who is good in mathematics can take the analytical view of anything and convert ideas into practical form by using mathematical models. There are a lot of applications that can be used for converting analog models into digital models by using mathematics. So all of these models require mathematical backgrounds and engineers are the people who bring ideas to reality. The same is applied on IT field too.

Answer:

step 3

Step-by-step explanation:

took test

The length of the garden is 20 feet and the width is 30 feet.

To solve this problem, we will use the formulas for the perimeter and area of a rectangle. The perimeter (P) of a rectangle is given by the formula P = 2(l + w), where l is the length and w is the width. The area (A) of a rectangle is given by the formula A = lw.

 Given the perimeter of the garden is 100 feet, we can set up the following equation:

[tex]\[ 2(l + w) = 100 \][/tex]

[tex]\[ l + w = 50 \][/tex]

[tex]\[ w = 50 - l \][/tex]

This equation relates the length and width of the garden.

 Next, we are given that the area of the garden is 600 square feet:

[tex]\[ lw = 600 \][/tex]

Substituting the expression for w from the first equation into the area equation, we get:

[tex]\[ l(50 - l) = 600 \][/tex]

[tex]\[ 50l - l^2 = 600 \][/tex]

Rearranging the terms, we have a quadratic equation:

[tex]\[ l^2 - 50l + 600 = 0 \][/tex]

To solve this quadratic equation, we can factor it:

[tex]\[ (l - 20)(l - 30) = 0 \][/tex]

This gives us two possible solutions for the length: l = 20 feet or l = 30 feet.

 Since we are asked to let the length be the shorter side, if necessary, we choose the smaller value for the length:

[tex]\[ l = 20 \text{ feet} \][/tex]

 Now we can find the width by substituting the length back into the equation for the perimeter:

[tex]\[ w = 50 - l \][/tex]

[tex]\[ w = 50 - 20 \][/tex]

[tex]\[ w = 30 \text{ feet} \][/tex]

 Therefore, the length of the garden is 20 feet and the width is 30 feet.

The length of the pond is 77.4 inches, and the width of the pond is 38.7 inches.

In geometry, a rectangular prism can be defined as a 3-dimensional solid shape that has six faces that are rectangles. A rectangular prism is also a cuboid.

Let,a be the length of the pond and b be the width of the pond.

The volume of the pond is;

[tex]\rm Volume \ of \ the \ pond= Area\ of\ the\ base\times deep\\\\Area\ of\ the\ base=\dfrac{Volume \ of \ the \ pond}{base}\\\\Area\ of\ the\ base=\dfrac{72000}{24}\\\\Area\ of\ the\ base=3000 in^2[/tex]

The length of the pond is two times its width.

a = 2b

The area of the base = a × b

3000 = ab

ab = 3000

Substitute the value of a in equation 2

[tex]\rm ab=3000\\\\2b \times b = 3000\\\\b^2=\dfrac{3000}{2}\\\\b^2=1500\\\\b=38.7[/tex]

Substitute the value of b in equation 1

[tex]\rm a=2b\\\\a=2 \times 38.7\\\\a=77.4[/tex]

Hence, the length of the pond is 77.4 inches, and the width of the pond is 38.7 inches.

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The width of the pond is approximately 38.7 inches and the length is approximately 77.4 inches.

To find the length and width of the pond, we can use the formula for the volume of a rectangular prism: length × width × depth. We are given that the depth is 24 inches and the volume is 72,000 cubic inches.

Let's denote the width as w.

Since the length is two times the width, we can express the width as w and the length as 2w.

Therefore, the equation for the volume becomes:

2w × w × 24 = 72,000

48w^2 = 72,000

w^2 = 1,500

w ≈ 38.7

Therefore, the width is approximately 38.7 inches and the length is approximately 77.4 inches.

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Answer with Step-by-step explanation:

We know that volume of a cylinder with radius r and height h is:

πr²h

and  volume of cone with radius r and height h is:

πr²h/3

Volume of cone=Volume of cylinder/3 < Volume of cylinder

Volume of cylinder=84π in² where π=3.14

Volume of cone=84π/3 in²

                          = 84(3.14)/3 in²

Hence, correct options are:

The volume of the cone is 1/3 the volume of the cylinder.

The volume of the cone is less than the volume of the cylinder.

The expression 84(3.14)/3 finds the volume of the cone.

the answer is: A, C, and D

Answer: The waitress works:[tex]\[{6 \frac{7}{8}}[/tex]   hours in the evening


Let us denote the number of hours the waitress works in the evening as E and the number of hours she works in the afternoon as A.

Given that:

[tex]\[A = 5 \frac{1}{8} \text{ hours}\][/tex]

We can convert the mixed number to an improper fraction:

[tex]\[5 \frac{1}{8} = 5 + \frac{1}{8} = \frac{40}{8} + \frac{1}{8} = \frac{41}{8}\][/tex]

So, [tex]\( A = \frac{41}{8} \)[/tex] hours.

The waitress works 1.75 hours less in the afternoon than in the evening. Thus, we have:

[tex]\[A = E - 1.75\][/tex]
[tex]\[\frac{41}{8} = E - 1.75\][/tex]

Converting 1.75 to a fraction:

[tex]\[1.75 = \frac{7}{4} = \frac{14}{8}\][/tex]

Now, the equation becomes:

[tex]\[\frac{41}{8} = E - \frac{14}{8}\][/tex]
[tex]\[E = \frac{41}{8} + \frac{14}{8}\]\[E = \frac{41 + 14}{8} = \frac{55}{8}\][/tex]

Convert the improper fraction back to a mixed number:

[tex]\[\frac{55}{8} = 6 \frac{7}{8}\][/tex]

Therefore, the waitress works:

[tex]\[{6 \frac{7}{8}}[/tex]   hours in the evening

The equation of the straight line whose x - intercept = 8 and

y - intercept = - 2 is y = x/4 - 2.

The slope - intercept form of a straight line is -

y = mx + c

Given is the equation of the line whose x-intercept is 8 and y-intercept is -2.

The given intercepts are -

x - intercept = 8

y - intercept = - 2

Then, the coordinates of the points through which the line passes are -

(0, -2) and (8, 0)

The slope of this line will be -

m = 0 - (- 2)/8 - 0

m = 2/8

m = 1/4

So the equation of the straight line will be -

y = x/4 - 2

Therefore, the equation of the straight line whose x - intercept = 8 and

y - intercept = - 2 is y = x/4 - 2.

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In a collision, both the car and truck experience the same force due to Newton's Third Law. However, due to Newton's Second Law, the car, with its smaller mass, experiences a greater acceleration.

According to Newton's Third Law of Motion, 'for every action, there is an equal and opposite reaction', both the truck and the car would experience the same magnitude of force. However, let's recall Newton's Second Law, 'force equals mass times acceleration' (F=ma). Using these physics principles, we can deduce that the car with the smaller mass would experience a greater acceleration, and the large truck would experience a lesser acceleration. This is because when the same force is applied to both objects, an object with a smaller mass will have a greater acceleration compared to an object with a larger mass. Hence, option B) 'The car has the greater acceleration because it has the smaller mass' would be the correct answer based on Newton's laws of motion.

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The equation that models this relation is y = 70/x. So, option (D) is the correct answer.

The constant of proportionality is the ratio of two proportional values at a constant value. Two variable values have a proportional relationship when either their ratio or their product gives a constant.

For the given situation,

The values are y = 10 and x = 7

y varies inversely with x.

⇒ [tex]y[/tex] ∝ [tex]\frac{1}{x}[/tex]

To remove proportionality we introduce a constant 'k'.

⇒ [tex]y=\frac{k}{x}[/tex]

Substitute the values of x and y,

⇒ [tex]10=\frac{k}{7}[/tex]

⇒ [tex]k=70[/tex]

Now substitute the value of k = 70 in y,

⇒ [tex]y = \frac{70}{x}[/tex]

Hence we can conclude that the equation that models this relation is      y = 70/x. So, option (D) is the correct answer.

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The number of hours if they worked the job together will be 2 hours.

A ratio is a collection of ordered integers a and b represented as a/b, with b never equaling zero. A proportionate expression is one in which two items are equal.

One maid can clean the house in 3 hours. Another maid can do the job in 6 hours.

We know that work and time are inversely proportional to each other.

Let T be the total time taken. Then the equation will be

1/T = 1/3 + 1/6

1/T = (2 + 1) / 6

1/T = 3 / 6

T = 6 / 3

T = 2

The number of hours if they worked the job together will be 2 hours.

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Final answer:

To find the side length of the smaller square, we take the square root of the larger square's area (400 sq inches) to get 20 inches, then halve it to align with the 1:2 scale factor of the sides for a correct length of 10 inches.

Explanation:

To solve this problem, we first acknowledge that the ratio of the areas of the two squares is 1:4. Since area is a function of the side length squared, we look for a scale factor of the side length that when squared will give us this ratio. In this case, the scale factor is 1:2, meaning the side length of the smaller square is half that of the larger square.

The area of the larger square is given as 400 square inches. To find the side length, we take the square root of the area, which is 20 inches. Therefore, the side length of the smaller square is half of this, which is 10 inches. However, given that the statement seems to expect an 8-inch side for the larger square, there might be a misunderstanding in the problem, assuming that area scales directly with side length rather than its square. Nonetheless, based on area and scale factor, the correct side length is 10 inches.

Answer:

90 degrees

Step-by-step explanation:

Answer:

-4-23i

Step-by-step explanation:

Above answer didn't quite understand what the question was asking. It's asking to SUBTRACT 8+7i from 4-16i. So it would be written like this:

(4-16i) - (8+7i).

Simplify it from there and you'd get -4-23i

To subtract 8 + 7i from 4 - 16i, subtract the real parts and the imaginary parts separately, resulting in -4 - 23i.

To subtract 8 + 7i from 4 − 16i, you treat the real and imaginary parts separately.

Subtract the real parts: 4 − 8 = −4.

Subtract the imaginary parts: (−16i) − (7i) = −16i − 7i = −23i.

Combine the results to get the final answer: −4 − 23i.

Therefore, subtracting 8 + 7i from 4 − 16i gives −4 − 23i.

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