Question 13
"s
the measure of one of the small angles of a right triangle is 30 less than 7 times
small angle. find the measure of both angles.
smallest angle:
other non-right angle:
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The duration of time period required is 2.4 years.

Compound interest simply means that the interest associated with a bank account, loan, or investment increases exponentially over time.

Given that, A person invests 3500 dollars in a bank. The bank pays 6.25% interest compounded annually.

A = P(1+r/100)ⁿ

n = time

A = amount

P = principal

11100 = 3500(1+0.0625)ⁿ

3.171 = (1.0625)ⁿ

Taking log to both sides,

log(3.171) = nlog(1.0625)

n = 2.40

Hence, the required time is 2.4 years.

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The sales price of the car, after a 20% discount, is $12,800, and the cost of the car including 13% tax is $14,464.

To determine the sale price of the car, we'll first calculate the discount amount by multiplying the regular price by the discount rate (20%).

Discount amount = Regular price × Discount rate

= $16000 × 0.20

= $3200

Next, we subtract the discount amount from the regular price to find the sale price.

Sale price = Regular price - Discount amount

= $16000 - $3200

= $12800

To calculate the cost of the car including tax, we need to add the tax amount to the sale price.

The tax rate is 13%.

Tax amount = Sale price × Tax rate

= $12800 × 0.13

= $1664

Cost of the car including tax = Sale price + Tax amount

= $12800 + $1664

= $14464

Therefore, the sales price of the car, after a 20% discount, is $12,800, and the cost of the car including 13% tax is $14,464.

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Answer:

x = 10.8

Step-by-step explanation:

9 ÷ x = x ÷ (9 + 4)

9 × (9 + 4) = x × x

9 × 13 = x²

117 = x²

x = 10.81665383

The explicit formula for f^-1 is (x-3)^(1/4) and this is obtained by switching the roles of x and y and solving for y in terms of x.

To find the inverse function of f(x)=x^4+3, we need to switch the roles of x and y, and solve for y.
Let y = x^4+3
Subtract 3 from both sides to get:
y - 3 = x^4
Take the fourth root of both sides to isolate x:
(x^4)^(1/4) = (y-3)^(1/4)
Simplify:
x = (y-3)^(1/4)
So the inverse function of f(x) is:
f^-1 (x) = (x-3)^(1/4)
This is the explicit formula for the inverse function of f(x).
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Eliminate \(y\).

\(u + v = (3x - 4y) + (x + 4y) = 4x \implies x = \dfrac{u+v}4\)

Eliminate \(x\).

\(u - 3v = (3x - 4y) - 3 (x + 4y) = -16y \implies y = \dfrac{3v-u}{16}\)

The Jacobian for this change of coordinates is

\(J = \begin{bmatrix} x_u & x_v \\ y_u & y_v \end{bmatrix} = \begin{bmatrix} \dfrac14 & \dfrac14 \\\\ -\dfrac1{16} & \dfrac3{16} \end{bmatrix}\)

with determinant

\(\det(J) = \dfrac14\cdot\dfrac3{16} - \dfrac14\cdot\left(-\dfrac1{16}\right) = \dfrac1{16}\)

The value of determinant of the matrix det (AB) is 15.

Given matrices A and B are 3 by 3 matrices with

det (A) = 3 and

det (b) = 5.

We need to find the value of det (AB).

Writing the given matrices into the augmented matrix form gives [A | I] and [B | I] respectively.

By multiplying A and B, we get AB. Similarly, by multiplying I and I, we get I. We can then write AB into an augmented matrix form as [AB | I].

Therefore, we can solve the augmented matrix [AB | I] by row reducing [A | I] and [B | I] simultaneously using elementary row operations as shown below.


The determinant of AB can be calculated as det(AB) = det(A) × det(B)

= 3 × 5

= 15.

Conclusion: The value of det (AB) is 15.

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We need to find the value of determinant det(AB), using the formula: det(AB) = det(A)det(B)

=> det(AB) = 3 × 5

=> det(AB) = 15.

Hence, the value of det(AB) is 15.

The given matrices are A and B. Here, we need to determine the value of det(AB). To calculate the determinant of the product of two matrices, we can follow this rule:

det(AB) = det(A)det(B).

Given that: det(A) = 3

det(B) = 5

Now, let C = AB be the matrix product. Then,

det(C) = det(AB).

To evaluate det(C), we have to compute C first. We can use the following method to solve the augmented matrix by elementary row operations.

Given matrices A and B are: Matrix A and B:

[A|B] = [3 0 0|1 0 1] [0 3 0|0 1 1] [0 0 3|1 1 0][A|B]

= [3 0 0|1 0 1] [0 3 0|0 1 1] [0 0 3|1 1 0].

We can see that the coefficient matrix is an identity matrix. So, we can directly evaluate the determinant of A to be 3.

det(A) = 3.

Therefore, det(AB) = det(A)det(B)

= 3 × 5

= 15.

Conclusion: Therefore, the value of det(AB) is 15.

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Answer:

The magnitude is sqrt((-4)^2 + (-9)^2) = 9.85. The angle is atan(-9/-4) = 180 deg + 66 deg = 246 deg = -114 deg.

Step-by-step explanation:

hope it help

Answer:

9.85

Step-by-step explanation:

|v|= √9²+(-4)²

=√81+16

=√97

|v|= 9.85

The solutions to the equation are x = -1, x = -8, and x = 1/2.

The equation 4x³ + 32x² + 42x - 16 = 0 can be divided throuh by 2 as follows:

2x³ + 16x² + 21x - 8 = 0

We can test each of these possible roots by substituting them into the equation and seeing if we get 0. When we substitute -1, we get 0, so -1 is a root of the equation. We can then factor out (x + 1) from the equation to get:

(x + 1)(2x² + 15x - 8) = 0

We can then factor the quadratic 2x² + 15x - 8 by grouping to get:

(x + 8)(2x - 1) = 0

This gives us two more roots, x = -8 and x = 1/2.

Therefore, the solutions to the equation are x = -1, x = -8, and x = 1/2.

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Answer:

I don't know it is actually pretty hard! But is this part of the rsm mcp contest

Step-by-step explanation:

The sum of digits of the products is given by S = 176

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is

Substituting the values in the equation , we get

The number 22 multiplied by a 22-digit number that contains only "2"s would result in a number that consists of a string of "4"s.

This is because 2 multiplied by 2 results in 4. Since all the digits in the 22-digit number are "2"s, each digit would multiply by 2 and result in "4".

Therefore, the product of 22 multiplied by a 22-digit number that contains only "2"s would be a 44-digit number consisting of "4"s.

The sum of the digits of a 44-digit number consisting of only "4"s can be calculated by multiplying the digit "4" by the total number of digits, which in this case is 44.

Sum of digits = 4 × 44 = 176

Hence , the sum of the digits of the product would be 176

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Answer:

\(48m^2-72mn\)

Step-by-step explanation:

We need to solve \((8m-3n)^2 - (4m+3n)^2\)

We know that,

\((a+b)^2=a^2+b^2+2ab\\\\(a-b)^2=a^2+b^2-2ab\)

Using the above formula,

\((8m-3n)^2 - (4m+3n)^2=(8m)^2+(3n)^2-2(8m)(3n)-[(4m)^2+(3n)^2+2(4m)(3n)]\\\\=64m^2+9n^2-48mn-(16m^2+9n^2+24mn)\\\\=64m^2+9n^2-48mn-16m^2-9n^2-24mn\\\\=48m^2-48mn-24mn\\\\=48m^2-72mn\)

So, the final answer is \(48m^2-72mn\).

we have

Write a quadratic equation in standard form whose graph passes through the given points (1,2) (2,-1) (3,-6)

A quadratic equation in standard form is equal to

y=Ax^2+Bx+C

substitute the given values

(1,2)

2=A(1^2)+B(1)+C

2=A+B+C ------> equation 1

(2,-1)

-1=A(2^2)+B(2)+C

-1=4A+2B+C -------> equation 2

(3,-6)

-6=A(3^2)+B(3)+C

-6=9A+3B+C -------> equation 3

step 1

In the equation 1 isolate the variable A

A=2-B-C -------> equation 4

step 2

substitute equation 4 in equation 2 and 3

-1=4(2-B-C)+2B+C

-1=8-4B-4C+2B+C

-1=8-2B-3C

2B+3C=

Answer:

12

Step-by-step explanation:

these are 2 similar triangles, as due to the construction we have 3 corresponding pairs of equal angles (the 47° angle is a given as equal, then due to the rules of crossing lines the inner angles as S must be equal as opposite angles, and that automatically makes the third angles equal, as the sum of all angles in a triangle must be always 180°).

in similar triangles the length ratio of corresponding sides is the same for all corresponding sides and lengths in these triangles.

so, we can use the sides opposite of the 47° angles to find that ratio :

21/9 = 7/3

and now also the sides 28 and x must have the same ratio :

28/x = 7/3

4/x = 1/3

12/x = 1

x = 12

the answer is d a decrease by 5/11

By taking the quotient between the total volume of paint and the volume needed to fill a container we can see that 16 containers can be filled.

We know that the art teacher has 12 + 4/5 gallons of paint, and we know that it requires 4/5 gallons to fill a single container.

Then the number of containers that can be filled is equal to the quotient between the total volume of paint and the volume needed fill a single container, it gives:

(12 + 4/5)/(4/5) = 16

So 16 containers can be totally filled.

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The triangle is obtuse, since the square of the longest side is greater than the sum of the squares of the other two sides.

The sum of the areas of the squares formed on the legs of the triangle equals the area of the square formed on the hypotenuse:

a²+b²=c²

a, b and c are side lengths

a=12,b=37 and c=40

12²+37²=40²

144+1369=1600

1513 is not equal to 1600

Since 1513 < 1600, we know that:

This means that the triangle is obtuse, since the square of the longest side is greater than the sum of the squares of the other two sides.

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Answer:

4/xt 3 sqaured

Step-by-step explanation:

(a) 732321847343 is not a valid UPC code.

(b) 726412175425 is a valid UPC code.

(c) 012345678903 is not a valid UPC code.

To determine whether a string of 12 digits is a valid UPC code, we need to check whether it satisfies the following conditions:

(a)  732321847343

This string has 12 digits, so it satisfies condition 1. However, the first digit is 7, which is not a valid number system digit.

Therefore, this is not a valid UPC code.

(b) 726412175425

This string has 12 digits, so it satisfies condition 1. The first digit is 7, which is a valid number system digit.

The next five digits (26412) are the manufacturer code, and the following five digits (17542) are the product code. The last digit (5) is the check digit.

Therefore, this is a valid UPC code.

(c) 012345678903

This string has 12 digits, so it satisfies condition 1. The first digit is 0, which is a valid number system digit.

However, the manufacturer code and product code (12345 and 67890, respectively) are not valid, as they do not correspond to any known manufacturer or product.

Therefore, this is not a valid UPC code.

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Answer:

Algebraic Expression.

Answer:

30/80

3/8x100

37.5

Step-by-step explanation:

Descriptive statistical analysis can be conducted on the data collected for computer repair service calls, and the worksheet "comprep5" contains information on the time taken for 30 service calls.

Descriptive statistical analysis involves the use of numerical and graphical methods to summarize and describe the main features of a data set. In the case of the computer repair service calls, the data collected on the time taken for 30 service calls can be analyzed using descriptive statistics to provide information on measures of central tendency, such as mean and median, as well as measures of variability, such as range and standard deviation.

The worksheet "comprep5" likely contains the raw data for the 30 service calls, which can be used to conduct the descriptive analysis.

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--The complete question is, What type of analysis can be conducted on the data collected for computer repair service calls, and what information is available in the worksheet entitled "comprep5"?--

Approximately 13.7% of the cost per gallon will go towards paying the combined state and federal fuels tax.

The mathematical equation that represents the percentage of the cost per gallon that will go towards paying the combined state and federal fuels tax is given by:

(Combined state and federal fuels tax / Cost per gallon of gasoline) * 100

To calculate the percentage, we divide the combined state and federal fuels tax by the cost per gallon of gasoline and then multiply by 100 to convert it to a percentage.

For example, if the cost per gallon of gasoline is $2.92 and the combined state and federal fuels tax is $0.40, we can substitute these values into the equation:

(0.40 / 2.92) * 100 = 13.7%

Therefore, approximately 13.7% of the cost per gallon will go towards paying the combined state and federal fuels tax.

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the explicit rule of the nth term of the arithmetic

sequence is A. aₙ = 6n-8

                    B. aₙ = 6n-26

                     C. aₙ= 6n-26

                     D. aₙ =6n-14

the sequence of numbers having constant differences between two successive numbers.

the explicit formula is used for determining any term of an arithmetic sequences directly in where only the first term and differences between two consecutive terms are needed to know.

Given, -20,-14,-8,-2,...are the terms. an explicit rule for the nth term of the arithmetic sequence is given by the formula, aₙ=a₁+(n-1) d

where, a₁ is the first term, aₙ is the nth term, d is the difference between two terms, n is the number of terms.

as per question a₁= -20, d= 6

now A. aₙ = -20+6(n+2)

           aₙ = 6n-8

B. aₙ= -20+6(n-1)

 aₙ = 6n-26

C. aₙ =-20+6(n-1)

   aₙ = 6n-26

D. aₙ= -20+6(n+1)

  aₙ = 6n-14

the above are the explicit rules for the arithmetic sequence.

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Answer:

(-1,1)

Step-by-step explanation:

To find where both lines intersect, solve for x and y with substitution.

Equation 1.

-8x+3y=11

Equation 2.

y=3x+4

Substitute y from equation 2 into equation 1.

-8x+3(3x+4)=11

-8x+9x+12=11

x=-1

Now plug x into the second equation to solve for y.

y=3(-1)+4

y=-3+4

y=1

Both lines intersect at (-1,1)

Answer:

  (-1, 1)

Step-by-step explanation:

It's somewhat difficult to find the point of intersection when graphing these by hand, but a graphing calculator can show it easily.

The lines intersect at (-1, 1).

__

You are given an expression for y, which makes it easy to use substitution as a solution method. Using the second equation to substitute for y in the first equation, we get ...

  -8x +3(3x +4) = 11

  x +12 = 11 . . . . simplify

  x = -1 . . . . . . . subtract 12

Substituting back into the second equation, we find y to be ...

  y = 3(-1) +4 = 1

The lines intersect at (x, y) = (-1, 1).

_____

Additional comment

The slopes are 8/3 and 3. When hand-drawn lines have nearly the same slope, it is difficult to tell exactly where they intersect. Of course, a graphing calculator does not have the problem of finite line width and an apparent long region of intersection. It can pinpoint the intersection exactly.

Of course, "substitution" is not the only way this system of equations can be solved. However, for this system, it is one of the easiest when doing the solution by hand.

Number of students need to be sampled = 610

A study to determine the true average number of alcoholic drinks consumed by Greek students is conducted.

Standard Deviation, σ = 6.3

Margin Error of population mean, E = 0.5

Probability = 95% = 0.95

The p-value corresponding to 0.95 probability = (1+0.95)/2 = 0.975

The z-score corresponding to this p-value = 1.96 [from the z-tables]

Now, The margin error, E = zσ/√n, where n is the sample size

⇒ n = (zσ/E)²

⇒ n = (1.96 x 6.3/0.5)²

⇒ n ≈ 610

Hence 610 students need to be sampled.

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Answer:

\(area = 4 \times ( \frac{1}{2} \times base \times height) \\ =2 \times ( \frac{1}{2} \times 16.3 \times 42.7) + 2 \times ( \frac{1}{2} \times 16.3 \times 22.5) \\ = 696.01 + 366.75 \\ = 1062.76 \\ = 1063\)

Therefore, the cylinders arranged in order from least volume to greatest volume are:

Cylinder with diameter 28 units and height 18 units

Cylinder with radius 13 units and height 17 units

Cylinder with radius 12 units and height 20 units

Cylinder with diameter 30 units and height 14 units

Volume is the measure of the amount of space occupied by a three-dimensional object. It is typically measured in cubic units, such as cubic meters or cubic feet. The formula for finding the volume of a solid depends on the shape of the object. For example, the volume of a cube can be found by multiplying the length, width, and height of the cube, while the volume of a sphere can be found using the formula (4/3)πr³, where r is the radius of the sphere.

Here,

To compare the volumes of the given cylinders, we need to use the formula for the volume of a cylinder, which is V = πr²h, where r is the radius of the circular base and h is the height of the cylinder. Let's calculate the volumes of the given cylinders:

Cylinder with diameter 28 units and height 18 units:

radius = 14 units

volume = π(14²)(18)

= 8,365.98 cubic units

Cylinder with radius 13 units and height 17 units:

volume = π(13²)(17)

= 8,724.68 cubic units

Cylinder with diameter 30 units and height 14 units:

radius = 15 units

volume = π(15²)(14)

= 9,420.98 cubic units

Cylinder with radius 12 units and height 20 units:

volume = π(12²)(20)

= 9,014.47 cubic units

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Step-by-step explanation:

x = -1, y = 4 We know that, the equation of perpendicular form is x/a + y/b = 1, -1/a + 4/b = 1, -1b + 4a/ab = 1, -1b + 4a = ab, 4a - 1b = ab

\( { - 8}x3 + {5x}^2 - 3x + 1\)

this is the standard form of the given algebraic expression

The equation of the line passing through M and perpendicular to PR would be y = -x + 6.

What is the equation of the line?

An equation of a line is a mathematical expression that describes the relationship between the variables x and y that defines the position of a point on the line. There are several ways to write the equation of a line, including slope-intercept form (y = mx + b), point-slope form (y - y1 = m(x - x1)), and standard form (ax + by = c).

The midpoint of a line segment is the point that is exactly halfway between the two endpoints of the line. To find the midpoint of a line segment, you need to take the average of the x-coordinates and the y-coordinates of the two endpoints.

Given the line segment Q(-6, 2) and R(10, 6), the midpoint is:

((-6+10)/2 , (2+6)/2) = (2,4)

So the midpoint of the line segment Q(-6, 2) and R(10, 6) is (2,4).

Now find the slope of PR = 6-(-4)/10 = 10/10=1

Hence, the slope of the line passing through M and perpendicular to PR would be = -1.

Now we can form the equation of the line:

y - 4 = -1(x - 2)

y - 4 = -x + 2

y = -x + 6

Hence, the equation of the line passing through M and perpendicular to PR would be y = -x + 6.

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