1. Select the ordered pair from the choices below that is a solution to the following system of equations:

4y = 2x + 10
8x − 3y = -14

a) (-1, 2)
b) (7, 6)
c) (-3, 2)
d) (5, 5)

2. Which of the following systems of equations has no solution?
a) 9x + 5y = 1; 15y = 18x − 4
b) -7x − 7 = 3y; -14y − 8 = -6x
c) 4x − 3y = 9; 6y = 8x − 18
d) 7y = 5x − 10; 10x − 14y = 8

3. Select the ordered pair from the choices below that satisfies following system of equations:

2x − 10y = -14
-4y = -x − 5

a) (-1/2,1)
b) (-20, 3)
c) (3, 2)
d) (8, -4)

To write the expression 4 log x − 6 log (x + 2) as a single logarithm, apply the properties of logarithms.

To write the expression 4 log x − 6 log (x + 2) as a single logarithm, we can make use of the following logarithmic properties:

Using these properties, we can simplify the given expression as follows:

4 log x − 6 log (x + 2) = log(x^4) - log((x + 2)^6)

Therefore, the expression can be written as the single logarithm log(x^4) - log((x + 2)^6).

The rate of the boat that travels 66 miles in 2 hours is approximately 2904 feet per minute.

Given:

Distance traveled = 66 miles

Time taken = 2 hours

We know that:

1 mile =  5280 feet

66 miles = 66 [tex]\times[/tex] 5280

                  348,480 feet

Next, we'll convert the time from hours to minutes.

 1 hour = 60 minutes,

2 hours = 120 minutes

Now, we can calculate the rate in feet per minute:

Rate (feet per minute) =  [tex]\dfrac{ Distance \ in\ feet}{Time\ in\ minutes}[/tex]

                    Rate             =        [tex]\dfrac{348480}{120}[/tex]

Rate   ≈ 2904 feet per minute

So, the rate of the boat is approximately 2904 feet per minute.

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Answer: Segment, line, ray

Step-by-step explanation: too answer was confusing and point was wrong.

The answers are a. ray, e. segment and f. line

In geometry, a one-dimensional figure has only length and no other dimensions such as width or height. Let's analyze each option:

Thus, the one-dimensional figures are the ray, segment, and line.

In the context of the number 1,711,799, the first 7 underlined is in the hundred thousands position, which gives it a value of seven hundred thousand (700,000).

In the given number 1,711,799, we're asked to find the value of the first underlined 7. Considering the location of the number, 7 is in the hundred thousands position. The value of a digit depends on its place in the number. Therefore, despite the fact that the digit is '7', because it is in the hundred thousands place, its value is seven hundred thousand (700,000).

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The first five non-zero multiples of 5 are 5, 10, 15, 20, and 25. These are obtained by multiplying the number 5 by the integers 1 to 5. All non-zero multiples of 5 will end with the digit 5 or 0, confirming their divisibility by 5.

Identifying Non-Zero Multiples of 5

When you're looking for non-zero multiples of 5, it's important to remember that the basic property of a multiple of 5 is that it ends in either 0 or 5. However, since we're focusing on non-zero multiples, we will exclude any multiples that end in zero. A multiple of a number is obtained by multiplying that number by an integer. For the number 5, the first five non-zero multiples are 5, 10, 15, 20, and 25. Notice that these numbers are the result of multiplying 5 by the integers 1 through 5, respectively.

Understanding that the concept of multiples relates to a number being multiplied by a sequence of whole numbers or integers gives us the ability to identify patterns within these numbers. For example, all non-zero multiples of 5 will end with the digit 5 or 0, reinforcing the idea that if a number ends in 5, it is divisible by 5 without remainder. This concept is supported by empirical observation and the understanding that patterns and numbers, such as these, can offer insight into broader mathematical ideas. Furthermore, the significance of the number 5 in various mathematical circumstances, such as the number of digits in a number or the properties of multiples, is apparent throughout different contexts.

Answer:

First of all, each slice represents an intersecting plane at that level.

For example, y = 2 is a plane that passes thorugh that level and cuts the volume given by [tex]x^{2} -y^{2}+z^{2}=2[/tex]

We replace this value in the given volume.

[tex]x^{2} -y^{2}+z^{2}=2\\x^{2} -(2)^{2}+z^{2}=2\\x^{2}+z^{2}=2+4\\x^{2}+z^{2}=6[/tex]

So, results in a circumference with radius [tex]\sqrt{6}[/tex], because a circumference is defind as [tex]x^{2} +y^{2} =r^{2}[/tex]. (On plane XY).

We repeat the process.

[tex](1)^{2} -y^{2}+z^{2}=2\\z^{2}-y^{2}=2-1\\z^{2}-y^{2}=1[/tex]

It forms a horizontal hyperbola on plane ZY.

[tex]x^{2} -0^{2}+z^{2}=2\\x^{2}+z^{2}=2[/tex]

Another circle with radius of [tex]\sqrt{2}[/tex] on plane XZ.

[tex](2)^{2} -y^{2}+z^{2}=2\\z^{2}-y^{2}=2-4\\z^{2}-y^{2}=-2\\\frac{z^{2}-y^{2}}{-2} =\frac{-2}{-2} \\\frac{y^{2} }{2} -\frac{z^{2} }{2} =1[/tex]

It forms a hyporbola on plane YZ.

Answer:

Option(B) is correct.

The tenth term is given by [tex]a_{10}=15+9(-2)[/tex]

Step-by-step explanation:

Given : The sequence 15, 13, 11, 9,....

We have to determine the equation that can be used to find the tenth term of the given sequence 15, 13, 11, 9,....

Consider the given sequence 15, 13, 11, 9,..

[tex]a_1= 15, a_2=13, a_3= 11[/tex]

Difference between each term is

[tex]a_2-a_1=13-15=-2\\ a_3-a_2=11-13=-2[/tex]

Since, The difference between each term is constant -2

Thus, The given sequence is an arithmetic sequence with first term 15 and common difference -2.

Thus, The general term is given by

[tex]a_n=a+(n-1)d[/tex]

where a is first term and d is common difference nd n is number of term.

Subsitute, we have,

[tex]a_{10}=15+(10-1)(-2)[/tex]

Simplify, we have,

[tex]a_{10}=15+9(-2)[/tex]

Thus, The tenth term is given by [tex]a_{10}=15+9(-2)[/tex]

Answer:

The scientific notation 2.15 × 10-3 has a value of

0.00215

Step-by-step explanation:

Step one

This problem bothers conversion from standard form otherwise known as scientific notation to decimal form

Step two

Now we are presented with the expression

2.15 × 10-³

To convert this it is equivalent to

2.15/10³= 2.15/1000

= 0.00215

On the other hand this can also be achieved without calculations by visually counting from the decimal point to the right 2 placements before inserting the first integer

This method saves alot of time expecialy when time is of essence.

Hence the scientific notation 2.15 × 10-3 has a value of

0.00215

Final answer:

To find the x-coordinates where the function f(x)=6sin(x)-sin^2(x) has a horizontal tangent line, we calculate the derivative, set it equal to zero, and solve for x, using trigonometric identities and potentially numerical methods. Thus, the solutions for \( x \) in the interval [tex]\( [0, 2\pi] \) are \( x = \frac{\pi}{2} + k\pi \), where \( k \)[/tex] is an integer. These solutions represent the points where [tex]\( \cos(x) = 0 \)[/tex], causing the equation to be satisfied.

Explanation:

The question asks to find the x-coordinates of all points on the graph of the function f(x)=6sin(x)-sin2(x) at which the tangent line is horizontal. To find these coordinates, we need to look for points where the derivative of the function is equal to zero since a horizontal tangent line has a slope of zero.

We start by finding the derivative of the function:

f'(x) = d/dx[6sin(x)-sin2(x)]

f'(x) = 6cos(x)-2sin(x)cos(x)

f'(x) = 6cos(x)-sin(2x) (using the double angle identity sin(2x)=2sin(x)cos(x))

To find when the derivative is zero, we set the derivative equal to zero:

0 = 6cos(x) - sin(2x)

We then need to solve this equation for x. This involves using trigonometric identities and potentially numerical methods if the equation cannot be solved analytically. The x-coordinates will be the solutions to this equation within the domain for which we are interested, typically 0 to 2π for one full cycle of a trigonometric function.

To solve the equation [tex]\( 0 = 6\cos(x) - \sin(2x) \)[/tex], we'll first try to rewrite it in terms of a single trigonometric function using known identities.

The double angle identity for sine states that [tex]\( \sin(2x) = 2\sin(x)\cos(x) \)[/tex]. Substituting this identity into the equation, we get:

[tex]\[ 0 = 6\cos(x) - 2\sin(x)\cos(x) \]\\Now, let's factor out \( \cos(x) \):\[ 0 = \cos(x)(6 - 2\sin(x)) \][/tex]

Now, for the equation to be true, either [tex]\( \cos(x) = 0 \) or \( 6 - 2\sin(x) = 0 \)[/tex].

1. If [tex]\( \cos(x) = 0 \), then \( x = \frac{\pi}{2} + k\pi \)[/tex], where \( k \) is an integer.

2. If [tex]\( 6 - 2\sin(x) = 0 \), then \( \sin(x) = 3 \)[/tex]. However, the sine function has a range of \([-1, 1]\), so there are no solutions for \( x \) in this case.

Thus, the solutions for \( x \) in the interval [tex]\( [0, 2\pi] \) are \( x = \frac{\pi}{2} + k\pi \), where \( k \)[/tex] is an integer. These solutions represent the points where [tex]\( \cos(x) = 0 \)[/tex], causing the equation to be satisfied.

The function t(n)=8n represents the number of tires needed for each truck. By substituting n=25 into the function, we find we need 200 tires for 25 trucks.

The function that represents the number of tires needed for each truck is t(n) = 8n. This means for each truck we need 8 tires, as n is the number of trucks and 8n signifies the total number of tires needed for those trucks.

If we want to find out the number of tires we would need for 25 trucks, we substitute n = 25 into the function, yielding t(25) = 8 * 25. When you perform this multiplication, the result is 200 tires.

So, for 25 trucks we would need 200 tires. Meaning "the correct answer is C. 200 tires".

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Answer:

The widest range lies between [68,100]

Step-by-step explanation:

Given is - to receive a grade of B in the class you must average exam score between 80% and 89% for all 4 exams including the final.

So, the range of the addition of all 4 scores is -

[tex]80*4=320[/tex]

[tex]89*4=356[/tex]

Now given scores addition is = [tex]83+85+84=252[/tex]

Now, the widest range of scores that you can get on the final exam in order to receive a grade of B can be found as :

[tex]320-252= 68[/tex]

[tex]356-252= 104[/tex]

So, the range lies between [68,100]