Three fractions that are equivalent to:
12/36
4/5 and
3/12 ...?

To find dy/dx of the equation 4x^3 - 3xy^2 + y^3 = 28 at the point (3,4), you can use implicit differentiation and solve for dy/dx.

To find dy/dx of the equation 4x^3 - 3xy^2 + y^3 = 28 at the point (3,4), we need to use implicit differentiation. We differentiate both sides of the equation with respect to x, treating y as a function of x. Thus, we get:

12x^2 - 3y^2 - 3x(2y(dy/dx)) + 3y^2(dy/dx) = 0

Now we substitute x = 3 and y = 4 into the equation and solve for dy/dx:

12(3)^2 - 3(4)^2 - 3(3)(2(4)(dy/dx)) + 3(4)^2(dy/dx) = 0

You calculate this by first understanding that '28 tens' means 280, and then dividing 280 by 4. So, 28 tens divided by 4 equals 70.

The phrase '28 tens' refers to 28 multiplied by 10, 28 x 10 = 280.

To find the result of dividing this by 4, we calculate 280 ÷  4 = 70.

Therefore, 28 tens divided by 4 equals 70.

The volume of the solid formed by rotating the region enclosed by y = x^3 and y = 9x about the x-axis in the first quadrant can be found by integrating from 0 to 3, the square of the outer and inner radius, multiplied by π. Solve the integral to get the volume.

To find the volume of the solid formed by rotating the region enclosed by y = x^3 and y = 9x about the x-axis, we need to use the method of discs/washers. The volume V is given by the following integral:

 ∫[a,b] π(r(x)^2 - R(x)^2) dx

For our given curves, r(x) = 9x and R(x) = x^3 since 9x ≥ x^3 for 0 ≤ x ≤ 3. Therefore, we get:

 V = ∫[0,3] π[(9x)^2 - (x^3)^2] dx

Solving this integral will yield the volume of the solid:

 V = π ∫[0,3] (81x^2 - x^6) dx

Calculating this integral will give the volume of the solid.

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Answer:

Option (b) is correct.

The general term of the sequence is 2n

Step-by-step explanation:

Given : The  sequence 2, 4, 6, 8, 10, . . .

We have to find the representation of the general term of the given sequence 2, 4, 6, 8, 10, . . .

Consider the given sequence 2, 4, 6, .....

The general term of an arithmetic sequence is given by [tex]a_n=a+(n-1)d[/tex]

where, a = first term

d is common difference

For the given sequence a = 2

and d = 2

Then [tex]a_n=2+(n-1)2=2+2n-2=2n[/tex]

Thus, The general term of the sequence is 2n

Answer with explanation:

Slope of Line= -2

The line passes through the point , (6,8).

⇒Equation of line passing through point , (a,b) having slope ,m is

y -b = m (x -a)

⇒≡Equation of line passing through point , (6,8) having slope ,-2 is

→y -8 = -2× (x -6)

→y -8 = -2 x + 12⇒⇒ Using Distributive property of multiplication with respect to Subtraction

→2 x + y= 12 + 8

→2 x + y=20

Required Equation of line.

Final answer:

To find the tangent lines to the curve y = (x - 1)/(x + 1) that are parallel to the given line, convert the given line to slope-intercept form to find the slope, take the derivative of the curve to find where its slope matches the line's slope, and utilize these points to write the equations of the tangent lines.

Explanation:

To find the equations of the tangent lines to the curve y = (x − 1)/(x + 1) that are parallel to the line x − 2y = 3, we first need to find the slope of the given line by rewriting it in slope-intercept form (y = mx + b), where m is the slope. Rewriting x − 2y = 3 gives us y = ⅓x - ⅓; thus, the slope (m) is ⅓.

Next, we find the derivative of the curve, y' = dy/dx, which will give us the slope of the tangent at any point x. Taking the derivative of y = (x − 1)/(x + 1) using the quotient rule or another differentiation method, we find a general expression for y'. We then set y' equal to ⅓ to find the points where the slope of the tangent is equal to the slope of the given line.

After determining the x-values where the tangent has the correct slope, we calculate the corresponding y-values on the curve and use these points to write the equations of the tangent lines in the form y = mx + b, substituting the slope (⅓) and our found points (x, y).

To find the equations of the tangent lines to the given curve that are parallel to the given line, we differentiate the curve's equation to find its slope, equate it to the slope of the given line, solve for x, substitute the values back into the curve's equation to find the corresponding y-values, and use the point-slope form of the equation of a line to find the equations of the tangent lines.

To find the equations of the tangent lines to the curve y = (x − 1)/(x + 1) that are parallel to the line x − 2y = 3, we can use the slope of the given line as the slope of the tangent lines. The slope of the given line is 1/2, so the slope of the tangent lines is also 1/2.

Next, we can differentiate the equation of the curve y = (x − 1)/(x + 1) with respect to x to find the slope of the curve at any point. Taking the derivative, we get dy/dx = 2/(x + 1)².

Since the tangent lines are parallel to the given line, their slopes are equal. Therefore, we can equate the slope of the curve to the slope of the tangent lines and solve for x:

2/(x + 1)² = 1/2

Solving this equation, we get x = -1 or x = 1.

Substituting these values of x back into the equation of the curve, we can find the corresponding y-values. The coordinates of the points where the tangent lines intersect the curve are (-1, -2) and (1, 2).

Finally, we can use the point-slope form of the equation of a line to find the equations of the tangent lines:

Tangent line at (-1, -2): y + 2 = (1/2)(x + 1)

Tangent line at (1, 2): y - 2 = (1/2)(x - 1)

Refer to the image attached.

Given: [tex]\angle BAC[/tex] and [tex]\angle BCA[/tex] are congruent.

To Prove: [tex]\Delta ABC[/tex] is an isosceles triangle.

Construction: Construct a perpendicular bisector from point B to line segment AC.  Label the point of intersection between this perpendicular bisector and line segment AC as point D.

Proof:

Consider [tex]\Delta BDA, \Delta BDC[/tex]

[tex]\angle BDA = \angle BDC= 90^\circ[/tex]

(By the definition of perpendicular bisector)

[tex]AD=DC[/tex] (By the definition of perpendicular bisector)

So, Line segment AD is congruent to DC by the definition of perpendicular bisector.

[tex]\angle BAC[/tex] = [tex]\angle BCA[/tex] (given)

So,  [tex]\Delta BDA \cong \Delta BDC[/tex] by ASA congruence postulate.

∆BAD is congruent to ∆BCD by the ASA congruence Postulate.

Line segment AB is congruent to line segment BC because corresponding parts of congruent triangles are congruent (CPCTC).

So, Option C is the correct answer.

Answer:

C.[tex]8.22 h\geq[/tex]$ 623

Step-by-step explanation:

We are given that Sophia is saving money for a new bicycle.

The bicycle will  cost atleast $623.

Sophia makes $8.22 per hour.

We have to find the inequality that could be used to find the number of hours Sophia needs to work to make enough money to buy a new bicycle.

Let Sophia works h hours to make enough money to buy a new bicycle.

Sophia makes money per hour =$8.22

Total money made by Sophia in h hours =[tex]8.22 h[/tex]

According to question

[tex]8.22 h\geq [/tex]$623

Hence, option C is true.

The digit in the tenths place of the number 123.456 is 4. In the general context of decimals and rounding, if the following digit (hundredths place) is 5 or higher, the tenths place is rounded up when dropped.

The number in the tenths place of 123.456 is 4. When looking at decimal numbers, the first digit to the right of the decimal point represents the tenths place. To illustrate, the number 123.456 can be broken down as (1  imes 10^2) + (2  imes 10^1) + (3  imes 10^0) + (4  imes 10^-1) + (5  imes 10^-2) + (6  imes 10^-3), where the digit 4 is in the tenths place and holds the value of four-tenths or 0.4.

Regarding rounding to the tenths place, if you had a number like 1,459.08 and need to round it, you would look at the digit in the hundredths place which is 8. Since the first dropped digit is 5 or higher, you round up, resulting in 1,459.1.

Final answer:

Deborah ran a total of 7 3/5 miles combined on Monday and Tuesday. This was determined by adding 3 2/5 miles and 4 1/5 miles together after converting them to improper fractions.

Explanation:

On Monday, Deborah ran 3 2/5 miles and on Tuesday she ran 4 1/5 miles. To find the total distance Deborah ran over these two days, we need to add the two distances together. When working with mixed numbers, it's often easiest to convert them to improper fractions before adding.

3 2/5 miles is the same as (3 × 5 + 2)/5 = 17/5 miles.
4 1/5 miles is the same as (4 × 5 + 1)/5 = 21/5 miles.

Adding these two fractions:

17/5 miles + 21/5 miles = (17 + 21)/5 miles

= 38/5 miles

= 7 3/5 miles

So, Deborah ran a total of 7 3/5 miles on Monday and Tuesday combined.

Answer:

therers no box

Step-by-step explanation:

The student is required to combine three algebraic fractions with different denominators using factoring to find a common denominator and then simplify the expression.

The question entails a topic in algebra, specifically with respect to combining expressions with different denominators, which requires finding a common denominator, and working with signs and exponents. The problem presents three fractions that should be combined: (x+6)/(x^2+8x+15), (3x)/(x+5), and (x-3)/(x+3).

Firstly, note that the denominator x^2+8x+15 can be factored into (x+3)(x+5). This will allow us to identify a common denominator for all three fractions, which is (x+3)(x+5). We rewrite the fractions so that each has the common denominator:

Now we simply combine these three fractions over the common denominator:

After the combination, the terms must be simplified and ordered in descending powers of x, which is the final answer.

Descending powers of x is [tex]\[ \frac{2x^2 + 8x + 15}{(x + 3)(x + 5)} \][/tex].

To combine the given expressions, we need to find a common denominator and then combine the numerators. Here are the expressions given:

[tex]\[ \frac{x}{x^2 + 8x + 15} + \frac{3x}{x + 5} - \frac{x - 3}{x + 3} \][/tex]

First, let's factor the quadratic denominator in the first term if possible and identify the common denominator:

The quadratic [tex]\( x^2 + 8x + 15 \)[/tex] can be factored into [tex]\( (x + 3)(x + 5) \)[/tex], since 3 and 5 are factors of 15 that add up to 8.

Now we have:

[tex]\[ \frac{x}{(x + 3)(x + 5)} + \frac{3x}{x + 5} - \frac{x - 3}{x + 3} \][/tex]

The common denominator will be [tex]\( (x + 3)(x + 5) \)[/tex].

Now, let's rewrite each fraction with the common denominator:

The second term already has [tex]\( x + 5 \)[/tex] in the denominator, so we multiply the numerator and denominator by [tex]\( x + 3 \)[/tex]to have the common denominator.

[tex]\[ \frac{3x}{x + 5} \rightarrow \frac{3x(x + 3)}{(x + 5)(x + 3)} \][/tex]

The third term has [tex]\( x + 3 \)[/tex] in the denominator, so we multiply the numerator and denominator by \( x + 5 \) to have the common denominator.

[tex]\[ \frac{x - 3}{x + 3} \rightarrow \frac{(x - 3)(x + 5)}{(x + 3)(x + 5)} \][/tex]

Now all terms have a common denominator, and we can combine them as follows:

[tex]\[ \frac{x}{(x + 3)(x + 5)} + \frac{3x(x + 3)}{(x + 3)(x + 5)} - \frac{(x - 3)(x + 5)}{(x + 3)(x + 5)} \][/tex]

Combine the numerators while keeping the denominator the same:

[tex]\[ \frac{x + 3x(x + 3) - (x - 3)(x + 5)}{(x + 3)(x + 5)} \][/tex]

Now, let's expand and simplify the numerator:

[tex]\[ x + 3x^2 + 9x - (x^2 + 2x - 15) \][/tex]

[tex]\[ x + 3x^2 + 9x - x^2 - 2x + 15 \][/tex]

Combine like terms:

[tex]\[ 3x^2 - x^2 + x + 9x - 2x + 15 \][/tex]

[tex]\[ 2x^2 + 8x + 15 \][/tex]

Now, let's put it all over the common denominator:

[tex]\[ \frac{2x^2 + 8x + 15}{(x + 3)(x + 5)} \][/tex]

This is the simplified expression in descending powers of \( x \). Since the numerator is already in descending powers of \( x \), this is the final answer. There is no further simplification possible because the numerator and the denominator do not have common factors other than 1.

The ratio that is not equivalent to 6:10 is 24/45, as it simplifies to 8/15 instead of 3/5 like the other options.

To determine which of the given ratios is not equivalent to 6:10, we can simplify the ratio 6:10 or convert it to a fraction and then reduce it to its simplest form. In fraction form, 6:10 can be written as 6/10, which simplifies to 3/5 when both the numerator and the denominator are divided by their greatest common divisor, which is 2.

Therefore, the ratio that is not equivalent to 6:10 is 24/45.

Final answer:

The values of m and n that satisfy the equation 15 - 28i = 3m + 4ni are m = 5 and n = -7 after equating and solving the real and imaginary parts separately.

Explanation:

To find the values of m and n that make the equation 15 - 28i = 3m + 4ni true, we must equate the real parts and the imaginary parts of both sides of the equation separately. The real part of the left side of the equation is 15, and the imaginary part is -28i. Similarly, for the right side, the real part is 3m, and the imaginary part is 4ni.

Equate the real parts: 15 = 3m
Dividing both sides by 3, we get m = 5.

Equate the imaginary parts: -28i = 4ni
Dividing both sides by 4i, we get n = -7.

Hence, the values that satisfy the equation are m = 5 and n = -7.