twin bothers, collin and cameron get jobs immediately after graduating from college at the age of 22. collin opts for the higher starting salary, $55,000 and stays with the same company until he retires at age 65. his salary doubles every 15 years. cameron opts for the lower starting salary, $35,000 but moves to a new job every few years so that he doubles his salary every 10 years until he retires at age 65. what is the annual salary of each brother upon retirement?

[tex]\bf \textit{Pythagorean Identities}\\\\ 1+tan^2(\theta)=sec^2(\theta) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ sec^2(\theta )cos^2(\theta )+tan^2(\theta )\implies \cfrac{1}{\begin{matrix} cos^2(\theta ) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix} }\cdot \begin{matrix} cos^2(\theta ) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix} +tan^2(\theta ) \\\\\\ 1+tan^2(\theta )\implies sec^2(\theta )[/tex]

Answer:

b. all real numbers

Step-by-step explanation:

The graphs of positive and negative x^2 parabolas will always have a domain of all real numbers.  Even though you only have a portion of the graph and see a "restriction" on your domain values, it is incorrect to assume that the domain is limited to what you can see.  As the branches of the parabola keep going up and up and up, the values of x keep getting bigger and bigger and bigger.  Again, this is true for all + or - parabolas.

Answer:

The error is C and the solution is B

Step-by-step explanation:

C is because she didn't look left to right to see that is goes on forever

B because you look left and right to see

Answer:

[tex]\large\boxed{(4x^2 + 8x - 2) - (2x^2 - 4x + 3)=2x^2+12x-5}[/tex]

Step-by-step explanation:

[tex](4x^2 + 8x - 2) - (2x^2 - 4x + 3)\\\\=4x^2 + 8x - 2 -2x^2 -(- 4x)- 3\\\\=4x^2+8x-2-2x^2+4x-3\qquad\text{combine like terms}\\\\=(4x^2-2x^2)+(8x+4x)+(-2-3)\\\\=2x^2+12x-5[/tex]

Answer:  The correct option is

(A) [tex]c+(c+2)=12,[/tex] where c is the number of dimes.

Step-by-step explanation:  Given that Alice has a total of 12 dimes and nickels and she has 2 more nickels than dimes.

We are to select the correct equation that represents the given problem situation.

Let c represents the number of dimes. Then, the number of nickels will be (c + 2).

Since there are total 12 coins, so the required equation is given by

[tex]c+(c+2)=12.[/tex]

Thus, the required equation is

[tex]c+(c+2)=12,[/tex] where c is the number of dimes.

Option (A) is CORRECT.

The expression for the total number of dimes and nickels is  c+(c+2)=12. Option A is the correct answer.

Given that Alice has a total of 12 dimes and nickels.

Also given that she has 2 more nickels than dimes.

Let us consider that c is the number of dimes. Then the number of nickels is given as,

Number of nickels = c+2

The total sum of dimes and nickels is 12, then,

Number of dimes + Number of nickels = 12

c + (c+2) = 12

Hence the expression for the total number of dimes and nickels is  c+(c+2)=12. Option A is the correct answer.

To know more about the sum, follow the link given below.

https://brainly.com/question/24412452.

11) Factors of 55 are 1,5,11,55 Factors of 75 are 1,3,5,15,25,75

The greatest common factor is 5.

12) With algebraic expressions you just simplify and multiplier in the simplification is the greatest common factor.

66yx + 30x^2y --) 6yx( 11 + 5x ) so the greatest common factor is 6yx.

13) 60y + 56x^2 --) 4( 15y + 14x^2 ) so the greatest common factor is 4.

14) 36xy^3 + 24y^2 --) 12y^2( 3xy + 2 ) so the greatest common factor is 12y^2.

15) 18y^2 + 54y^2 --) 18y^2( 1 + 3 ) so the greatest common factor is 18y^2.

16) 80x^3 + 30yx^2 --) 10x^2( 8x + 3y ) so the greatest common factor is 10x^2.

17) 105x + 30yx + 75x --) 15x( 7 + 2y + 5 ) so the greatest common factor is 15x.

18) 140n + 140m^2 + 80m --) 20( 7n + 7m^2 + 4m ) so the greatest common factor is 20.

If you want a further explanation step by step just ask :)

Hello there!

Answer:

$24

Step-by-step explanation:

In order to find the answer to your problem, we're going to need to find out how much ONE liter of paint costs.

Lets gather the information of what we know:

2 1/2 liters of paint

↑ Cost $60.

With the information we know, we can solve to find the answer.

In order to get the answer, we would need to divide 60 by 2 1/2 (or 2.5). We would need to do this because when we divide it, it would allow us to get the cost for 1 liter.

Lets solve:

[tex]60 \div 2.5=24[/tex]

When you divide, you should get the answer of 24.

This means that one liter of paint cost $24.

$24 should be your FINAL answer.

Answer:

$24/liter

Step-by-step explanation:

Write the the dollar amount first and the paint volume second in this ratio:

 $60.00

--------------- = $24/liter

2.5 liters

Answer:

To the nearest tenth, the height of the balloon is 2.9 miles

Step-by-step explanation:

The nearer point takes the greater angle of elevation.

The diagram is shown in the attachment.

The height of the balloon above the ground is c unit.

From triangle ABD,

[tex]\tan 42\degree=\frac{c}{x}[/tex]

[tex]\implies x=\frac{c}{\tan 42\degree}[/tex]...eqn1

From triangle ABC,

[tex]\tan 20\degree=\frac{c}{x+4.8}[/tex]

[tex]\implies x+4.8=\frac{c}{\tan 20\degree}[/tex]

[tex]\implies x=\frac{c}{\tan 20\degree}-4.8[/tex]..eqn2

We equate both equations and solve for c.

[tex]\frac{c}{\tan 42\degree}=\frac{c}{\tan 20\degree}-4.8[/tex]

[tex]\frac{c}{\tan 42\degree}-\frac{c}{\tan 20\degree}=-4.8[/tex]

[tex]\implies (\frac{1}{\tan 42\degree}-\frac{1}{\tan 20\degree})c=-4.8[/tex]

[tex]\implies -1.636864905c=-4.8[/tex]

[tex]\implies c=\frac{-4.8}{-1.636864905}[/tex]

[tex]c=2.932435039[/tex]

To the nearest tenth, the height of the balloon is 2.9 miles

Answer:

on usatestprep its 1.2

Step-by-step explanation:

Answer:

The determinant of A (the main matrix) is -3; the determinant of y is 30; the determinant of x is 18; the solution to the system is (-6, -10)

Step-by-step explanation:

Set up the matrix to find the determinant of the main matrix.  Find the determinant by multiplying the numbers on the major axis and subtract from that the multiplication of the numbers on the minor axis:

[tex]\left[\begin{array}{ccc}2&-1&\\1&-2\\\end{array}\right][/tex]

Find the determinant by multiplication:

(2×-2)-(1×-1)= -3

To find the determinant of y, replace the second column with the solutions to have a matrix that looks like this:

[tex]\left[\begin{array}{ccc}2&-2\\1&14\\\end{array}\right][/tex]

To find the determinant of that matrix by multiplication:

(2×14)- (1× -2) = 30

Lastly, find the determinant of x by replacing the first column with the solutions.  That matrix will look like this:

[tex]\left[\begin{array}{ccc}-2&-1\\14&-2\\\end{array}\right][/tex]

Find the determinant of x by multiplication:

(-2 × -2) - (14 × -1) = 18

Now we want Cramer's Rule that tells us if we divide the determinant of [tex]A_{x}[/tex]

by the determinant of A, we will find the value of x:

[tex]\frac{A_{x} }{A}=\frac{18}{-3}  =-6[/tex]

and the same for y:

[tex]\frac{A_{y} }{A}=\frac{30}{-3}=-10[/tex]

So the solution to the system is (-6, -10)

2x - y = -2

x = 14 + 2y

2x - y = -2

x - 2y = 14

The system determinant = -3

2 (-2) - 1 (-1)

-4 + 1

-3

The y-determinant = 30

(14) - 1 (-2)

28 + 2

30

The x-determinant = 18

-2 (-2) - 14 (-1)

4 + 14

18

The solution is x = -6 and y = -10 or (-6,-10)

x = 18/-3

x = -6

y = 30/-3

y = -10

f(x) = 7 sin (4πx) + 3. The function f(x) = 7 sin (4πx) + 3 describe a sinusoidal function whose period is 1/2, maximum value 10, minimum value -4, and it has a y-intercept of 3.

A sinudoidal function whose period is 1/2, maximum value is 10, minimum value is -4, and it has a y-intercept of 3. Let's write to the form f(x) = A sin (ωx +φ) + k, where A is the amplitude, ω is the angular velocity with ω=2πf, (ωx+φ) is the oscillation phase, φ the initial phase (horizontal shift), and k is y-intercept (vertical shift).

Calculating the amplitude:

A = |max - min/2|

A = |10 - (-4)/2| = 14/2

A = 7

calculating the ω:

The period of a sinusoidal is T = 1/f --------> f = 1 / T

ω = 2πf -------> ω = 2π ( 1/T) with T = 1/2

ω = 2π (1/(1/2) = 2π (2)

ω = 4π

The y-intercept k = 3

Writing the equation function with A = 7, ω = 4π, k = 3, φ = 0.

f(x) = A f(x) = A sin (ωx +φ) + k ----------> f(x) = 7 sin (4πx) + 3.

Answer:

Probability of the sticks weight being 2.44 oz or greater is 0.01017 .

Step-by-step explanation:

We are given that the manufacturer's website states that the average weight of each stick is 2.00 oz with a standard deviation of 0.19 oz.

Also, it is given that the weight of the drumsticks is normally distributed.

Let X = weight of the drumsticks, so X ~ N([tex]\mu = 2,\sigma^{2} = 0.19^{2}[/tex])

The standard normal z distribution is given by;

               Z = [tex]\frac{X-\mu}{\sigma}[/tex] ~ N(0,1)

Now, probability of the sticks weight being 2.44 oz or greater = P(X >= 2.44)

P(X >= 2.44) = P( [tex]\frac{X-\mu}{\sigma}[/tex] >= [tex]\frac{2.44-2}{0.19}[/tex] ) = P(Z >= 2.32) = 1 - P(Z < 2.32)

                                                     = 1 - 0.98983 = 0.01017

Therefore, the probability of the sticks weight being 2.44 oz or greater is 0.01017 .

Final answer:

The probability of the sticks weighing 2.44 oz or more is approximately 0.01017.

Explanation:

Given that the manufacturer's website states that the average weight of each stick is 2.00 oz with a standard deviation of 0.19 oz, we know the weight of the drumsticks is normally distributed.

Let X represent the weight of the drumsticks, with X being normally distributed with a mean (μ) of 2 and a variance  [tex](\sigma^2) \ of \ 0.19^2[/tex]

To find the probability of the sticks weighing 2.44 oz or more, we need to calculate P(X ≥ 2.44).

We can standardize X using the formula Z = (X - μ) / σ, which results in a standard normal distribution with mean 0 and standard deviation 1.

So, to find P(X ≥ 2.44), we compute P((X - μ) / σ ≥ (2.44 - 2) / 0.19), which simplifies to P(Z ≥ 2.32).

From the standard normal distribution table or a calculator, we find that P(Z < 2.32) is approximately 0.98983.

Therefore, P(Z ≥ 2.32) = 1 - P(Z < 2.32) = 1 - 0.98983 = 0.01017.

Hence, the probability of the sticks weighing 2.44 oz or more is approximately 0.01017.

That would be p times 1.97 or 1.97p (letter D) this is because each pound is worth 1.97 dollars more so if you bought 1 pound of rice you'd pay only $1.97 but if you bought 5 pounds of rice you'd pay $9.85 since 1.97 times 5 is 9.85

Hope this helped!

Let me know if this helped!

Answer:

67.0 cm^3

Step-by-step explanation:

The volume of the cylinder is given by the formula ...

V = πr^2·h

The volume of the hemisphere is given by the formula ...

V = (2/3)πr^3

The volume of the two figures together will be ...

V = πr^2·h + (2/3)πr^3 = πr^2(h +2/3r)

V = π(2 cm)^2·(4 cm + 2/3·2 cm) = 64π/3 cm^3

V ≈ 67.0 cm^3

Try this options:

a. total - 6 digits, '6' - 1 digit, then probability of rolling a '6' is 1/6;

b. total - 6 digits, '6' - 1 digit, then probability of rolling 1,2,3,4,5 is 5/6;

c. if probability of rolling a '6' is p and not rolling a '6' is q, then p+q=1;

d. if expected probability of one rolling a '6' is 1/6, then numbers of times of rolling a '6' during 120 times is 120/6=20 times.

Answer:

C: 48 straws

Step-by-step explanation:

First, find the perimeter of one picture frame: (2 x length) + (2 x width).  Convert 20 centimeters to millimeters so that you are working in the same units; there are 10 millimeters in 1 centimeter, so 20 centimeters = 200 millimeters.  

(2 x 200) + (2 x 120) = 400 + 240 = 640

Each picture frame has a perimeter of 640 millimeters.

Next, figure out how many straws are needed for one picture frame:

640/80 = 8

Jana uses 8 straws for each picture frame.  Since she is decorating 6 picture frames, solve 6 x 8 = 48.  

Jana needs 48 straws to complete her project.

Answer:

a)  333,135,504 different plates

b) 230,315,904 different plates

c) 180,835,200 different plates

Step-by-step explanation:

Pattern: Digit(1-9)-Letter-Letter-Letter-Letter-Digit(1-9)-Digit (1-9)

We will calculate the number of possibilities for the digits part, then for the letters part, then we'll multiply them together.

For the digits, we have 3 numbers, first and last 2 positions. We can consider this is a single 3-digit number, where n = 9 (since they are non-zero digits) and r = 3.  

For the letters part, it's basically a 4-letter word, where n = 26 (A through Z) and r = 4.

(a) How many different license plate numbers are possible?

No limitation on repeats for this question:

For the digits, we have 9 * 9 * 9 = 729 (since repetition is allowed, and we can pick any digit from 0 to 9 for each position)

For the letters we have: 26 * 26 * 26 * 26 = 456,976

Because the digits and letters arrangements are independent from each other, we multiply the two numbers of possibilities to have the global number of possibilities:

P = 729 * 456976 = 333,135,504 different plates, when there's no repeat limitation.

(b) How man license plate numbers are possible if no digit appears more than once?

Repeats limitation on digits:

For the digits, we have 9 * 8 * 7 = 504 (since repetition is NOT allowed, we can pick any of 9 digits for first position, then any 8 remaining and finally any 7 remaining at the end)

For the letters we still have: 26 * 26 * 26 * 26 = 456,976

Because the digits and letters arrangements are independent from each other, we multiply the two numbers of possibilities to have the global number of possibilities:

P = 504 * 456976 = 230,315,904 different plates, when there's no repeat on the digits.

(c) How man license plate numbers are possible if no digit or letter appears more than once?

Repeats limitation on both digits and letters:

For the digits, we have 9 * 8 * 7 = 504 (

For the letters we still have: 26 * 25 * 24 * 23 = 358,800

Because the digits and letters arrangements are independent from each other, we multiply the two numbers of possibilities to have the global number of possibilities:

P = 504 * 358800 = 180,835,200 different plates, when there's no repeat on the digits AND on the letters.

Answer:

The son's age is 10 and the daughter's age is 5 now

Step-by-step explanation:

Let

x-----> the son's age now

y----> the daughter's age now

we know that

x=2y ----> equation A

(x+15)+(y+15)+(40+15)=100

x+y+85=100

x+y=15 -----> equation B

Substitute equation A in equation B and solve for y

2y+y=15

3y=15

y=5 years

Find the value of x

x=2(5)=10 years

therefore

The son's age is 10.

The daughter's age is 5

Answer:

165 ways  to choose the paintings

Step-by-step explanation:

This is clearly a Combination problem since we are selecting a few items from a group of items and the order in which we chosen the items does not matter.

The number of possible ways to choose the paintings is;

11C3 = C(11,3) = 165

C denotes the combination function. The above can be read as 11 choose 3 . The above can simply be evaluated using any modern calculator.

Answer:

165 ways

Step-by-step explanation:

Total number of painting, n = 11

Now, three of them are chosen randomly to display in the gallery window.

Hence, r = 3

Since, order doesn't matter, hence we apply the combination.

Therefore, number of ways in which 3 paintings are chosen from 11 paintings is given by

[tex]^{11}C_3[/tex]

Formula for combination is [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]

Using this formula, we have

[tex]^{11}C_3\\\\=\frac{11!}{3!8!}\\\\=\frac{8!\times9\times10\times11}{3!8!}\\\\=\frac{9\times10\times11}{6}\\\\=165[/tex]

Therefore, total number of ways = 165

Answer:

(0,3) and (3,0)

Step-by-step explanation:

The first thing to do is graph the two equations to see where they intersect. Then you know what answer to look for. The graph is below. It was done on desmos.

I take it the first equation is a typo and should be y = x^2 - 4x + 3

Equate the two equations.

-x + 3 = x^2 -4x + 3    Subtract 3 from both sides

-x = x^2 - 4x + 3-3

-x = x^2 - 4x                Add x to both sides.

0 = x^2 - 4x + x

0 = x^2 - 3x                Factor

0 = x(x - 3)

So x can equal 0

or x can equal 3

In either case the right side will reduce to 0.

Case 1. x = 0

y= - x + 3

y = 0 + 3

y = 3

So the point is (0,3)

Case 2. x = 3

y = - x + 3

y = - 3 + 3

y = 0

So the point is (3,0)

Answer:

Step-by-step explanation:

This is a negative x^2 quadratic.  I'm not sure if there's anything else you need.

1. 10^5

2. z^5*x^3*y^2

3. 1

[tex]5 \div 100 \times 399 \times 1 = 19.95 \: and \: 4 \div 100 \times 399 \times 1 = 15.96[/tex]

Answer:

x = 20

Step-by-step explanation:

Formula

x1/x2 = x3/x4

Givens

x = 11

x2 = 11 + 121 = 132

x3 = 10

x4 = 10 + 5x + 10

Solution

11/132 = 10 / (5x + 10 + 10)        Combine

11/132 = 10/(5x + 20)                Cross multiply

11*(5x + 20) = 132 * 10              Combine on the right.

11(5x + 20 ) = 1320                   Divide by 11. (You could remove the brackets, but this is easier.

11(5x + 20)/11 = 1320/11            Do the division

5x + 20 = 120                          Subtract 20 from both sides

5x + 20-20 = 120 - 20             Combine

5x = 100                                   Divide by 5

5x/5 = 100/5

x = 20

The answer would be c.

[tex] \sqrt{28} [/tex]

Answer:

[tex]f(-1)=5[/tex]

Step-by-step explanation:

We know that the equation is

[tex]f(x)=-4x^3+2x^2-1[/tex]

We can then plug -1 in for x

[tex]f(-1)=-4(-1)^3+2(-1)^2-1\\\\f(-1)=-4(-1)+2(1)-1\\\\f(-1)=4+2-1\\\\f(-1)=5[/tex]

ANSWER

[tex]f( - 1) = 5[/tex]

EXPLANATION

The given function is

[tex]f(x) = - 4 {x}^{3} + 2 {x}^{2} - 1[/tex]

We substitute x=-1 to obtain:

[tex]f( - 1) = - 4 {( - 1)}^{3} + 2 {( - 1)}^{2} - 1[/tex]

We simplify to obtain;

[tex]f( - 1) = 4 + 2 - 1[/tex]

.

This evaluates to

[tex]f( - 1) = 5[/tex]

Answer:

The y-intercept is -2

The equation of the axis of symmetry is x = 1 ⇒ 3rd answer

Step-by-step explanation:

* Lets revise the general form of the quadratic function

- The general form of the quadratic function is f(x) = ax² + bx + c,

 where a, b , c are constant

# a is the coefficient of x²

# b is the coefficient of x

# c is the y-intercept

- The meaning of y-intercept is the graph of the function intersects

 the y-axis at point (0 , c)

- The axis of symmetry of the function is a vertical line

  (parallel to the y-axis) and passing through the vertex of the curve

- We can find the vertex (h , k) of the curve from a and b, where

 h is the x-coordinate of the vertex and k is the y-coordinate of it

# h = -b/a and k = f(h)

- The equation of any vertical line is x = constant

- The axis of symmetry of the quadratic function passing through

  the vertex then its equation is x = h

* Now lets solve the problem

∵ f(x) = -2x² + 4x - 2

∴ a = -2 , b = 4 , c = -2

∵ The y-intercept is c

∴ The y-intercept is -2

∵ h = -b/2a

∴ h = -4/2(-2) = -4/-4 = 1

∴ The equation of the axis of symmetry is x = 1

Answer:

Step-by-step explanation:

f(x) is an easy one.  Because it's a parabola, it's standard form is

[tex]y=ax^2+bx+c[/tex]

But even simpler than that, look at a point on the graph, in particular, (2, 4).  If x = 2 and y = 4, we can square 2 to get 4, so the equation for that is the parent graph, [tex]y=x^2[/tex], plain and simple.

The next one requires a bit of doing.  Pick 3 points on the graph because we have 3 unknowns to find: a, b, and c.  The points that are easy to pick are (0, -2), (2, -4), (-2, -4).  Use the x and y coordinates from each one of those points to fill in the standard form of the parabola.  Because this parabola is "upside down" the leading coefficient is negative.  Start with the first coordinate first:

[tex](0, -2)-->-2=-a(0)^2+b(0)+c[/tex] which gives us that c = -2.  That's good...one down, 2 to go.

Next we will use the remaining 2 points to create a system of equations that we can solve simultaneously for a and b.  Using the second coordinate pair (2, -4):

[tex]-4=-a(2)^2+b(2)-2[/tex] gives us the simplified equation:

***-2 = -4a + 2b***

I put the stars in front and behind because we will need to come back to that one in a minute.

Using the last coordinate pair (-2, -4):

[tex]-4=-a(-2)^2-b(2)-2[/tex] simplifies down to:

***-2 = -4a - 2b***

Now put these together and solve the system by elimination, and you see that 2b and the -2b cancel each other out, leaving you with -4 = -8a, so a = 1/2.  Now we know a:  1/2 and c:  -2 and we can find b:

If -2 = -4a + 2b, then -2 = -4(1/2) + 2b, and b = 0.  That means that the equation for the upside down parabola is

[tex]y=-\frac{1}{2}x^2-2[/tex]

Answer: First Option

−2 • f(x)

Step-by-step explanation:

The function [tex]y=log_2(x)[/tex] passes through point (2,1) since the exponential function [tex]2 ^ x = 2[/tex] when [tex]x = 1[/tex].

Then, if the transformed function passes through the point (2, -2) then this means that f(x) was multiplied by a factor of -2. So if an ordered pair [tex](x_0, y_0)[/tex] belonged to f(x), then [tex](x_0, -2y_0)[/tex] belongs to the transformed function. Therefore, if [tex]f(x) = log_2 (x)[/tex] passed through point (2, 1) then the transformed function passes through point (2, -2)

The transformation that multiplies to f(x) by a factor of -2 is:

[tex]y = -2 * f (x)[/tex]

and the transformed function is:

[tex]y = -2log_2 (x)[/tex]

Answer:

24 cm², 44 cm², and 66 cm²

Step-by-step explanation:

The rectangular prism has six faces.  The opposite faces have the same area, so we can say there are three faces with unique areas.

The face on the bottom of the rectangular prism has an area of:

A = 11 cm * 4 cm = 44 cm²

The face on the side of the rectangular prism has an area of:

A = 4 cm * 6 cm = 24 cm²

And the face on the front of the rectangular prism has an area of:

A = 11 cm * 6 cm = 66 cm²

So 24 cm², 44 cm², and 66 cm² are all answers that apply.

Answer:

1 5/12 inches.

Step-by-step explanation:

That is 3 1/6 - 1 3/4

= 19/6 - 7/4

The lowest common denominator of 4 and 6 is 12, so we have:

38/12 - 21/12

= 17 /12

= 1 5/12 inches (answer).

The remaining string length after cutting [tex]\(1 \frac{3}{4}\)[/tex] inches is [tex](1 \frac{5}{16}\)[/tex]inches.

The correct option is (a).

find out how much string is left when [tex]\(1 \frac{3}{4}\)[/tex] inches are cut from a piece initially measuring[tex]\(3 \frac{1}{16}\)[/tex]inches.

1. Convert the mixed numbers to improper fractions:

[tex]- \(1 \frac{3}{4}\) inches = \(\frac{7}{4}\) inches[/tex]

[tex]- \(3 \frac{1}{16}\) inches = \(\frac{49}{16}\) inches[/tex]

2. Make the denominators equal:

  - Multiply the numerator and denominator of [tex]\(\frac{7}{4}\)[/tex]by 16 to make the denominators equal:

[tex]\(\frac{7}{4} = \frac{112}{64}\)[/tex]

  - Now we have:

    - Initial length = [tex]\(\frac{49}{16}\)[/tex] inches

    - Cut length = [tex]\(\frac{112}{64}\)[/tex] inches

3. Subtract the two fractions:

  - Subtract the cut length from the initial length:

[tex]\(\frac{49}{16} - \frac{112}{64}\)[/tex]

  - To subtract, we need a common denominator. The least common multiple (LCM) of 16 and 64 is 64.

 - Convert both fractions to have a denominator of 64:

[tex]\(\frac{49}{16} = \frac{196}{64}\)[/tex]

[tex]\(\frac{112}{64}\) remains the same.[/tex]

- Subtract the numerators:

   [tex]\(\frac{196}{64} - \frac{112}{64} = \frac{84}{64}\)[/tex]

4. Simplify the result:

  - Divide both the numerator and denominator by their greatest common factor (GCF), which is 4:

[tex]\(\frac{84}{64} = \frac{21}{16}\)[/tex]

5. Convert back to a mixed number:

  - Divide the numerator by the denominator:

[tex]\(\frac{21}{16} = 1 \frac{5}{16}\)[/tex]

Therefore, the remaining string length after cutting [tex]\(1 \frac{3}{4}\)[/tex] inches is [tex](1 \frac{5}{16}\)[/tex]inches.

ANSWER

191 is closest to nearest whole number.

EXPLANATION

Let the distance from the top of the tree to the tip of the shadow be l feet as shown in the diagram.

This is the same as the hypotenuse of the right triangle.

The given side length, 130 ft is adjacent to the angle of elevation which is 47°

We use the cosine ratio to obtain,

[tex] \cos(47 \degree)= \frac{adjacent}{hypotenuse} [/tex]

[tex]\cos(47 \degree)= \frac{130ft}{l} [/tex]

[tex]l= \frac{130ft}{\cos(47 \degree)} [/tex]

[tex]l =190.6162941[/tex]

The distance from the top of the tree to the tip of the shadow, given the length of the shadow is 130 feet and the angle of elevation is 47 degrees, is approximately 180 feet.

The question is asking us to find the distance from the top of the tree to the tip of the shadow using given information: the shadow cast by the tree is 130 feet long, and the angle of elevation is 47 degrees. For this, we can use the tangent function in trigonometry, which is defined as the opposite side over the adjacent side in a right triangle. Here, the length of the shadow (130 feet) serves as the adjacent side, and the height of the tree serves as the opposite side.

To find the hypotenuse (the distance from the top of the tree to the tip of the shadow), you can use the formula: Hypotenuse = Adjacent / cos(angle). So, the Hypotenuse = 130 feet / cos(47) = approximately 180 feet.

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