A computer’s hard disk is spinning at 90 revolutions per second. Through how many degrees does it travel in a second? in a minute?

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.

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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Answer:

D. z+nm

Step-by-step explanation:

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

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.

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

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:

 [tex]log_{b} b^{n}[/tex] = n.

Step-by-step explanation:

Given :  [tex]log_{b} b^{n}[/tex] , b is any positive number not equal to 1 and n is any number.

To find : Evaluate the expression  [tex]log_{b} b^{n}[/tex].

Formula used : [tex]log_{b} x^{y}[/tex]. = y ∙ [tex]log_{b}[/tex](x) and

[tex]log_{b}(b) = 1.

Solution : We have  [tex]log_{b} b^{n}[/tex].

By logarithm rule :  [tex]log_{b} x^{y}[/tex]. = y ∙ [tex]log_{b}[/tex](x).

Then  [tex]log_{b} b^{n}[/tex] = n∙ [tex]log_{b}[/tex](b).

By logarithm rule :  [tex]log_{b}(b) = 1.

Now, n∙ [tex]log_{b}[/tex](b) = n.

Therefore,  [tex]log_{b} b^{n}[/tex] = n.

Answer:

1. B

2. D

3. B

4. A

Step-by-step explanation:

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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).

The third angle for the triangle is equal to 89°.

Geometry is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer.

A triangle is a three-edged polygon with three vertices. It is a fundamental shape in geometry. Triangle ABC denotes a triangle with vertices A, B, and C.

Given that the two angles of a triangle measure 43 degrees and 48 degrees. the value of the third angle will be calculated as,

Third angle = 180 - 43 - 48

Third angle = 89°

The complete question is given below.

If two angles of a triangle measure 43 degrees and 48 degrees what is the third angle of the triangle?

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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.

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

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: 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.