Which test point holds true for y − 2x ≤ 1?

(0, 2)

(-2, 4)

(1, 4)

(5, 0)

Answer: D

Step-by-step explanation:

The correct transformation of the equation is achieved by completing the square to obtain the form (x + 5)^2 = 1, which gives the values of p = -5 and q = 1.

The transformation of the quadratic equation x^2 + 10x + 24 = 0 into the form (x - p)^2 = q involves completing the square. To complete the square, one must find a number that, when added to x^2 + 10x, completes a perfect square trinomial. This number is calculated by taking half of the x-coefficient (b/2), squaring it, and adding it to both sides of the equation. In this case, (10/2)^2 = 25. We then have x^2 + 10x + 25 on the left and 25 - 24 on the right, which simplifies to (x + 5)^2 = 1. The values of p and q are thus -5 and 1, respectively.

Answer:

1. B

2. D

3. B

4. A

Step-by-step explanation:

To find the point on the sphere at which you are looking in the direction of (-2, -3, 2), we need to find the intersection of the line passing through (2, 3, 0) and the direction (-2, -3, 2) with the sphere of radius 1 centered at the origin. By substituting the parametric equations of the line into the equation of the sphere and solving for t, we can find the two points on the sphere where you will be looking.

To find the point on the sphere at which you look when you are looking in the direction of (-2, -3, 2), we need to find the intersection of the line passing through (2, 3, 0) and the direction (-2, -3, 2) with the sphere of radius 1 centered at the origin.

The equation of the line passing through (2, 3, 0) and in the direction (-2, -3, 2) can be expressed as:

x = 2 - 2t

y = 3 - 3t

z = 2t

Substitute these equations into the equation of the sphere, we get:

(2 - 2t)2 + (3 - 3t)2 + (2t)2 = 1

Simplifying the equation:

17t2 - 14t + 2 = 0

Using the quadratic formula:

t = (-b ± √(b2 - 4ac))/(2a)

Substituting the values, we get two possible values of t: t ≈ 0.23 and t ≈ 0.1.

Substituting these values back into the parametric equations of the line:

For t ≈ 0.23:

x ≈ 1.54

y ≈ 1.31

z ≈ 0.46

For t ≈ 0.1:

x ≈ 1.8

y ≈ 2.1

z ≈ 0.2

So, the points on the sphere at which you look when you are looking in the direction of (-2, -3, 2) are approximately (1.54, 1.31, 0.46) and (1.8, 2.1, 0.2).

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To minimize the weight of the tank, the base length should be (4/3) times the height. The height should be (3/4)^(1/4) times the fourth root of (2V/π).

To find the dimensions that will make the tank weight as little as possible, we need to consider the volume and weight of the tank. Since the tank is square-based, the volume can be calculated by V = bh^2, where b is the length of one side of the base and h is the height.

The weight of the tank can be calculated by multiplying the volume by the density of the stainless steel. W = V * density. Since we want to minimize the weight, we want to minimize the volume.

To do this, we can differentiate the volume equation with respect to b and set it equal to zero. Solving for b, we find that b = (4/3) * h. Substituting this value into the volume equation gives: V = (4/3) * h^3.

To further minimize the volume, we can take the derivative of the volume equation with respect to h and set it equal to zero. Solving for h, we find that h = (3/4)^(1/4) * (2V/π)^(1/4).

Boys and girls, we love to teach. The students love it. Teaches a student is to be grateful (pleasing). We love to teach the disciples of the history of the Roman letter ("by means of Literature", abl. Of means). In the game a lot (many) are students. * * To mature to learn the use of Latin in the disciples

Answer: 30°, 300° and 330°

Step-by-step explanation:

This is a quadratic equation in trigonometry format.

Given 2 sin^2 θ − sin θ − 1 = 0

Let a constant 'k' = sin θ...(1)

The equation becomes

2k²-k-1 =0

Factorizing the equation completely we have,

(2k²-2k)+(k-1) = 0

2k(k-1)+1(k-1)=0

(2k+1)(k-1)=0

2k+1=0 and k-1=0

2k = -1 and k=1

k=-1/2 and 1

Substituting the value of k into equation 1 to get θ

sin θ = 1

θ = arcsin1

θ = 90°

Similarly

sin θ = -1/2

θ = arcsin-1/2

θ = -30°

This angle is negative and falls in the 3rd and 4th quadrant

In the third quadrant, θ = 270 +30 = 300° and

in the 4th quadrant, θ = 360 - 30° = 330°

Therefore the values of θ are 30°, 300° and 330°

I hope you find this helpful?

Final answer:

The trigonometric equation is transformed into a quadratic equation by substituting sin θ with x. Then, the quadratic formula is applied to find x, which is then used to find the solutions for θ in radians, considering the periodic nature of the sine function. The solutions are then verified.

Explanation:

To solve the trigonometric equation 2 sin^2 θ − sin θ − 1 = 0, treat it as a quadratic equation by setting x = sin θ. The reformed equation is 2x^2 - x - 1 = 0. Now, factor this equation or use the quadratic formula to find the values of x, and subsequently the values of θ.

Using the quadratic formula:

Therefore, the solutions for x are:

Convert these back into solutions for θ by finding θ such that sin θ = x. Use units of radians for angles and remember to consider the periodic nature of the sine function.

Answer:

Check if the answers are reasonable by substituting back into the original equation and verifying that they produce true statements.

Final answer:

The largest earthquakes recorded in Chile, Russia, and Alaska measured 9.5, 9.1, and 9.2 on the Richter scale respectively, with Chile's earthquake being the largest and Russia's the smallest.

Explanation:

The three largest earthquakes ever recorded took place in Chile, Russia, and Alaska. Based on the information provided, the earthquake in Chile was the largest, measuring 9.5 on the Richter scale. The earthquake in Alaska, being larger than the one in Russia but not as large as the one in Chile, would measure 9 1/5, or 9.2, on the Richter scale. Lastly, the earthquake in Russia was the smallest of the three, measuring 9 1/10, or 9.1, on the Richter scale.

These seismic events, marked by their enormity on the Richter scale, underscore the geological dynamics in different regions of the world. The specific magnitudes assigned to each earthquake provide a quantitative measure of their intensity, offering valuable insights for understanding and preparing for such natural phenomena.

The list of 5 numbers with 6 in the tens place are [tex]\boxed{265,{\text{ 67, 163, 969, 2165, 31462}}}.[/tex]

Further explanation:

Explanation:

The whole numbers is the series of numbers that starts from zero.

The natural numbers are those numbers that start from one.

The place values are only natural numbers.

The base ten systems represent the position of a place value.

After decimal the first place is the tenth place, the second place is the hundredth and the third place is the thousandth.

Consider a number as [tex]265.[/tex]

Here, number 5 is on the ones place, number 6 is on the tens place and 2 is on the hundreds place.

Hence, the list of 5 numbers with 6 in the tens place are [tex]\boxed{265,{\text{ 67, 163, 969, 2165, 31462}}}.[/tex]

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Answer details:

Grade: Middle School

Subject: Mathematics

Chapter: Decimals

Keywords: list of 5 numbers, numbers, hundreds place, tens place,  whole number, decimal, compare, contrast, methods, base ten system, place value, decimal expansion, natural numbers, real numbers.

Final answer:

Five numbers with 6 in the tens place are 60, 61, 62, 63, and 64. Each has the digit 6 immediately to the left of the ones place.

Explanation:

To list 5 numbers with 6 in the tens place means that each number must have its second digit from the right as a 6. Here are five numbers that meet this criterion:

60

61

62

63

64

Each of these numbers has a 6 in the tens place, which we can confirm by looking at the digit immediately to the left of the ones place.

Answer: 3 and 1/5

Step-by-step explanation: To write an improper fraction as a mixed number, divide the denominator into the numerator.

[tex]\sqrt[5]{16}[/tex] = 3 with a remainder of 1

Therefore, 14/5 can be written as the mixed number 3 and 1/5.

Answer:

10,000

Step-by-step explanation:

2 x 5= 10 then add 3 more zeros

The shortest distance [tex]x + y = 4[/tex] that is closest to the origin is [tex]\boxed{2\sqrt 2 {\text{ units}}}.[/tex]

Further explanation:

The formula for distance between the two points can be expressed as follows,

[tex]\boxed{{\text{Distance}} = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} }[/tex]

Given:

The line is [tex]x + y = 4.[/tex]

Explanation:

The coordinate of the origin is [tex]\left( {0,0} \right).[/tex]

The first point is [tex]\left( {x,y} \right)[/tex] and the second point is [tex]\left( {0,0} \right).[/tex]

The distance between the two points can be calculated as follows,

[tex]\begin{aligned}{\text{Distance}}&= \sqrt {{{\left( {x - 0} \right)}^2} + {{\left( {y - 0} \right)}^2}}\\&= \sqrt {{x^2} + {{\left( {4 - x} \right)}^2}}\\&= \sqrt {{x^2} + {x^2} - 8x + 16}\\&= \sqrt {2{x^2} - 8x + 16}\\\end{aligned}[/tex]

Differentiate the above equation with respect to [tex]x[/tex].

Substitute the first derivative equal to zero.

[tex]\begin{aligned}\frac{d}{{dx}}\left( {{\text{Distance}}} \right) &= 0\\\frac{{\left( {2x - 4} \right)}}{{\sqrt {2{x^2} - 8x + 16} }} &= 0\\2x - 4 &= 0\\2x &= 4\\x&= 2\\\end{aligned}[/tex]

Substitute [tex]x = 2[/tex] in equation [tex]x + y = 4[/tex] to obtain the value of [tex]y[/tex].

[tex]\begin{aligned}2 + y &= 4\\y&= 4 - 2\\y&= 2\\\end{aligned}[/tex]

The point is [tex]\left( {2,2} \right).[/tex]

The shortest distance can be obtained as follows,

[tex]\begin{aligned}{\text{Distance}} &= \sqrt {{2^2} + {2^2}}\\&= \sqrt {4 + 4}\\&= 2\sqrt 2\\\end{aligned}[/tex]

The shortest distance [tex]x + y = 4[/tex] that is closest to the origin is [tex]\boxed{2\sqrt 2 {\text{ units}}}.[/tex]

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Application of derivatives

Keywords: Derivative, shortest distance, curve, origin, attains, maximum, value of x, function, differentiate, minimum value, closest point, line, y+x = 4.

The correct answer is:

The conversion factor would be the number of blocks multiplied by the weight of 1 block.

Explanation:

Consider the following example:

Suppose 1 block weighs 3 pounds. If there are 15 blocks, to find the total weight of the blocks, we multiply 3 by 15: 3(15) = 45. This means the total weight would be 45 pounds.

For any given number of blocks and any given weight of 1 block, to find the total weight of the blocks we would multiply the number of blocks by the weight of 1 block.

Answer:

D. z+nm

Step-by-step explanation:

Equaivalent to mn+z: Just switch the letters around and you get z+nm

To find the coordinate pair in the solution set for y < 1 - 6x, substitute x and y values into the inequality and check if it is satisfied.

The question asks for a coordinate pair that is in the solution set for the inequality y < 1 - 6x. To find the solution set, we need to find the values of x and y that satisfy the inequality. Let's pick a few coordinate pairs and substitute the values into the inequality to see if they satisfy it.

For example, let's try the coordinate pair (1, 0). Substituting x = 1 and y = 0 into the inequality, we get 0 < 1 - 6(1), which simplifies to 0 < 1 - 6, and further simplifies to 0 < -5. Since this statement is false, the coordinate pair (1, 0) is not in the solution set.

Similarly, we can try other coordinate pairs and substitute the values into the inequality to check if they satisfy it. The coordinate pair that satisfies the inequality will be in the solution set.

Answer:

Both will have same inertia.

Step-by-step explanation:

The inertia of a body is defined as its property to oppose the state of rest or uniform motion. In this question, we have to write out of four kilograms of iron and four kilograms of cork which will have maximum inertia. Inertia of a body is also equal to measure of its mass.

In this case, the masses of both iron and cork are same i.e. 4 kg. So, it is clear that the inertia of both iron and cork are same due to their same masses.

Answer:

0.10x + 0.15(50 – x) = 0.12(50)

Step-by-step explanation:

Answer C. If two chords are the exact same distance from the center of the circle then those two chords must be congruent.