Simplify the expression

Answer:

Step-by-step explanation:

After caculating, a) Matrix is given below, b) C₁ = 65460/193,C₂ = 48480/193, C₃ = 64460/193, c)The mass input to reactor 3 is reduced by 40g/day and d) Change of concentration is Δ C₃ = 2950/193

To solve the system of equations, we can use Gaussian elimination or any other suitable method. Here, we will use Gaussian elimination:

Given:

15C₁-2C₂-C₃=4000

-3C₁+18C₂-6C₃=1500

-4C₁-C₂+12C₃=2400

Therefore:

[tex]A=\left[\begin{array}{ccc}15&-3&-1\\-3&18&-6\\-4&-1&12\end{array}\right][/tex][tex]B=\left[\begin{array}{ccc}4000\\1500\\2400\end{array}\right][/tex]

a) The inverse matrix of A would be

[tex]A^-^1\left[\begin{array}{ccc}14/193&37/2895&12/965\\4/193&176/2895&31/965\\5/193&9/965&87/965\end{array}\right][/tex]

b) For solution of [C] = [A^-1][B]

[tex]C=\left[\begin{array}{ccc}C1\\C2\\C3\end{array}\right] =\left[\begin{array}{ccc}14/193&37/2895&12/965\\4/193&176/2895&31/965\\5/193&9/965&87/965\end{array}\right] \\[/tex]

C₁ = 65460/193

C₂ = 48480/193

C₃ = 64460/193

c) For increasing the concentration [C₁] by 10

C₁ = 65460/193 +10

=67390/193

Now again,

[tex]\left[\begin{array}{ccc}15&-3&-1\\-3&18&-6\\-4&-1&12\end{array}\right] \left[\begin{array}{ccc}67390/193\\48480/193\\64460/193\end{array}\right] =\left[\begin{array}{ccc}4150\\1470\\2360\end{array}\right][/tex]

So from the calculation= 2360 -2400= -40

Therefore the mass input to reactor 3 is reduced by 40g/day

d) Now again reducing mass input

[tex]B=\left[\begin{array}{ccc}4000-500\\1500-250\\2400\end{array}\right] = \left[\begin{array}{ccc}3500\\1250\\2400\end{array}\right] \\\\C=\left[\begin{array}{ccc}C1\\C2\\C3\end{array}\right] = \left[\begin{array}{ccc}173530/579\\130640/579\\61510/193\end{array}\right][/tex]

Δ C₃=C₃-C₃’

=64460-61510/193

= 2950/193

Δ C₃ = 2950/193

Therefore that is the change of concentration.

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The median number of TVs in Mrs. Jamison's class is 2, (i.e. the middle number).

To find the median of the given data of the number of TVs in Mrs. Jamison's class:

First we have to list all the values from least to greatest.

Then, find the middle number (in case if there is one middle number) then that will be the median of the data.

And if there is two middle number, we have to find the average of the two middle numbers and the result will be our answer i.e. median.

So, The data is 0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,6

The middle number from the data is 2

                        Therefore, the median is 2.

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The gross income for selling 348 bushels of apples at $16.50 per bushel is $5,732.

The formula to calculate gross income from the sale of a certain number of bushels of apples at a certain price per bushel is Gross Income = Number of Bushels x Price per Bushel. In this case, the calculation of the gross income for selling 348 bushels of apples at $16.50 per bushel would be Gross Income = 348 Bushels x $16.50/Bushel = $5,732.

To calculate the gross income, first we must multiply the number of bushels of apples (348) by the price per bushel ($16.50). When multiplied, the numbers result in $5,732. This means that, when selling 348 bushels of apples at $16.50 per bushel, the gross income would be $5,732.

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The length of the intercepted arc is (35 - 14√2 ) units.

what is length?

Length is a physical quantity that describes the distance between two points. It is typically measured in units such as meters, centimeters, inches, or feet. In mathematics, length can refer to the size of a geometric object,

In the given question,

We know that the length of an arc of a circle is given by the formula L = r*theta, where r is the radius of the circle and theta is the angle in radians subtended by the arc at the center of the circle.

Here, the radius of the circle is 7 units and the angle subtended by the arc is 5π/4 radians. Therefore, the length of the intercepted arc is:

L = 7*(5π/4) = (35π/4) units.

To express the answer in simplest form, we need to rationalize the denominator. Multiplying both the numerator and denominator by 2√2, we get:

L = (35π/4)(2√2/2√2)

= (35*π*√2)/(8) units.

Finally, simplifying this expression, we get:

L = (35 - 14√2) units.

Therefore, the length of the intercepted arc is (35 - 14√2) units

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The area of the composite figure is 96. 522 cm²

First, we need to know that the formula for the area of a triangle is expressed as;

Area = 1/2 × base × height

Now, substitute the values, we have;

Area = 1/2 × 6 × 8

Multiply the values

Area = 1/ 2 × 48

Divide the values

Area = 25 cm²

The area of the sector is represented as;

Area = θ/360 πr²

substitute the values

Area = 82/360 × 3.14 × 10²

Area = 71. 522 cm²

The total area of the figure = 25 + 71. 522 = 96. 522 cm²

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Answer:

y = 1x-5

Step-by-step explanation:

See attached worksheet.

We'll look for a line with the form y=mx+b, where m is the slope and y is the y-intercept.

Pick any two points on the line, but pick ones that are clearly on known lines so that the points are more accurate.  Pick one at the x=o point, if it can be read clearly.  The vaue of y at x=0 is the y-intercept.

Follow the steps in the attachement to find the equation of the line, which is

  y=1x-5

Lοzanο's paid a tοtal οf $460.25 fοr the bοat rental, including sales tax and tip.

In mathematics, percentages are a way οf expressing a number as a fractiοn οf 100. It is οften used tο cοmpare values, represent prοpοrtiοns, and calculate changes οr discοunts.

The cοst οf the bοat rental fοr the day was $350. The sales tax at a rate οf 6.5% is calculated as:

Sales tax = 6.5% οf $350 = 0.065 × $350 = $22.75

Sο the tοtal cοst οf the bοat rental with sales tax is:

Tοtal cοst = $350 + $22.75 = $372.75

The Lοzanο's alsο decided tο tip their guide 25% οf the οriginal cοst οf the bοat rental, which is:

Tip = 25% οf $350 = 0.25 × $350 = $87.50

Therefοre, the tοtal amοunt they paid is:

Tοtal amοunt = Tοtal cοst + Tip = $372.75 + $87.50 = $460.25Sο the Lοzanο's paid a tοtal οf $460.25 fοr the bοat rental, including sales tax and tip.

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Answer: $460.25 for the boat rental, including sales tax and tip.

Step-by-step explanation The cost of the boat rental for the day was $350. The sales tax at a rate of 6.5% is calculated as:

Sales tax = 6.5% of $350 = 0.065 × $350 = $22.75

So the total cost of the boat rental with sales tax is:

Total cost = $350 + $22.75 = $372.75

The Lozano's also decided to tip their guide 25% of the original cost of the boat rental, which is:

Tip = 25% of $350 = 0.25 × $350 = $87.50

Therefore, the total amount they paid is:

Total amount = Total cost + Tip = $372.75 + $87.50 = $460.25

So the Lozano's paid a total of $460.25 for the boat rental, including sales tax and tip.

The real and imaginary values of w is 86  and - 0.92i respectively.

To find w = √(3 - 4i), we can use the following steps:

Step 1: Find the modulus and argument of z = 3 - 4i

The modulus of z is

|z| = √(3² + (-4)²)

= √(9 + 16) = √25

= 5.

The argument of z is arg(z) = arctan(-4/3) ≈ -0.93 radians (or about -53.13 degrees).

Step 2: Find the principal square root of the modulus of z, which is

√|z| = √5.

Step 3: Find the argument of w, which is half of the argument of z, i.e., arg(w) = arg(z)/2

= -0.93/2

≈ -0.465 radians (or about -26.57 degrees).

Step 4: Express w in terms of its real and imaginary parts, using the formula:

w = √|z| * exp(i*arg(w)).

Substituting the values we found above, we get:

w = √5 x exp(i(-0.465))

= √5 x (cos(-0.465) + i*sin(-0.465))

≈ 1.86 - 0.92i

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Answer:

4 or 5

Step-by-step explanation:

because rational numbers also include natural numbers but natural numbers starts from 1 to eternity

while whole numbers start from 0 and is also included in rational numbers so it might be 5

f(x) = (8/9)(x-3)² - 4 which has a vertex at (3,-4) and passes through the point (0,4).

What is a quadratic equation?

A quadratic equation is a second-degree polynomial equation of the form:

ax² + bx + c = 0

where a, b, and c are constants, and x is the variable. Quadratic equations can have one, two, or zero real solutions, depending on the values of the constants a, b, and c. The solutions can be found using the quadratic formula:

x = (-b ± [tex]\sqrt{b^2 - 4ac}[/tex]) / 2a or by factoring the quadratic expression into two linear factors.

A quadratic function can be expressed in the form:

[tex]$$f(x) = a(x-h)^2 + k$$[/tex]

where (h,k) is the vertex of the parabola.

From the problem, we have the vertex (h,k) = (3,-4). Substituting these values into the equation gives:

[tex]$$f(x) = a(x-3)^2 - 4$$[/tex]

To find the value of a, we use the fact that the function passes through the point (0,4). Substituting x=0 and y=4 into the equation gives:

[tex]$$4 = a(0-3)^2 - 4$$[/tex]

Simplifying and solving for a, we get:

a=8/9

Therefore, the quadratic function is:

f(x) = (8/9)(x-3)² - 4

which has a vertex at (3,-4) and passes through the point (0,4).

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The variable "Annual income payments: $186,928" is of the type "Numerical, continuous, ratio."

The given information that Robert filled on his application for a loan is as follows:Annual income payments: $186,928Number of credit cards: 4Ever convicted of a felony: NoOwn a second car: NoThe types of variables are:Numerical: A numerical variable is a measurable quantity, and they are of two types: discrete and continuous.Discrete variables: They are quantitative variables that can be counted and are usually whole numbers. In this case, there is no discrete numerical variable, so it is not a valid option.Continuous variables: They are quantitative variables that can be measured and are usually decimal. In this case, the only continuous numerical variable is the annual income payments which amounts to $186,928.Categorical: A categorical variable is a variable that is defined by how the categories can be defined. It does not have a numerical meaning, unlike a numerical variable. It is further divided into two types: nominal and ordinal.Nominal variables: They are variables that have no order or ranking. In this case, the ever convicted of a felony is a nominal variable.Ordinal variables: They are variables that have an order or ranking. In this case, there is no ordinal categorical variable, so it is not a valid option.Therefore, the variable "Annual income payments: $186,928" is of the type "Numerical, continuous, ratio." The variable "Ever convicted of a felony" is of the type "Categorical, nominal."

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Overall, there are infinitely many possible combinations of angle measures that satisfy the equation x + (75.3 - x) = 75.3.

It is true that 75.3 = 75.3 when the equation x + (75.3 - x) = 75.3 is reduced to its simplest form. This means that any value of x that falls within the range of possible angle measurements, which is 0 to 75.3 degrees, will satisfy the equation.

As a result, the number of possible combinations of angle measurements that fulfil the equation is unlimited. The particular values of each angle can vary, but each combination will consist of two angles whose measures total up to 75.3 degrees.

For instance, if the sum of the two angles is 75.3 degrees, one conceivable angle combination is 30 degrees and 45.3 degrees. The combination of 60 degrees and 15.3 degrees is another conceivable one.

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The value of f(0) for the given function is 1÷2.

In mathematics, a function is a relation between two sets of values, where each value in the first set (called the domain) is associated with exactly one value in the second set (called the range). A function can be represented by a formula or an equation, which specifies how the input values are transformed into output values.

To find the value of f(0) for the given function f(x) = 1/2sin(x÷3-90)+1, we need to substitute 0 for x in the expression for f(x) and simplify:

f(0) = 1÷2sin(x÷3-90)+1

f(0) = 1÷2sin(-90)+1

f(0) = 1÷2(-1)+1 [since sin(-90) = -1]

f(0) = -1÷2 + 1

f(0) = 1÷2

Therefore, the value of f(0) for the given function is 1÷2.

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Complete question:

Find the value of F(0) when F(x)= 1/2sin(0/3-90)+1.

So the scale factor of triangle ABC to triangle DEF is 2/5.

Similar triangles are two triangles that have the same shape, but not necessarily the same size. This means that their corresponding angles are equal, and their corresponding sides are proportional.

Here,

To verify if the two triangles are similar, we need to check if their corresponding angles are congruent and if their corresponding sides are proportional.

Checking corresponding angles:

∠A corresponds to ∠D

∠B corresponds to ∠E

∠C corresponds to ∠F

Checking corresponding sides:

AB/DE = 10/25 = 2/5

BC/EF = 16/40 = 2/5

CA/FD = 20/50 = 2/5

Since the corresponding angles are congruent and the corresponding sides are proportional, we can conclude that triangle ABC is similar to triangle DEF. To find the scale factor of triangle ABC to triangle DEF, we can take any corresponding side and divide it by the corresponding side of the other triangle. For example:

AB/DE = 2/5

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The inequality -1/2 ≤ ab ≤ 1/2 implies [tex](ab)^2[/tex] ≤ 1/4, and squaring bοth sides οf the inequality gives us  [tex](ab)^2[/tex] ≤ 1/4 as required.

Tο prοve that -1/2 ≤ ab ≤ 1/2 fοr  [tex]a^2+b^2 = 1[/tex]and a, b ∈ ℝ, we can start by nοting that:

-1 ≤ a ≤ 1 (because [tex]a^2 \le a^2 + b^2 = 1[/tex], sο -1 ≤ a ≤ 1)

-1 ≤ b ≤ 1 (because [tex]b^2 \le a^2 + b^2 = 1[/tex], sο -1 ≤ b ≤ 1)

Multiplying these inequalities, we get:

-1 ≤ ab ≤ 1

Nοw, we need tο shοw that ab cannοt equal ±1. If ab = 1, then we have:

[tex]a^2 + b^2 = 1[/tex]

[tex]a^2 + 2ab + b^2 = 1 + 2ab[/tex]

[tex](a + b)^2 = 1 + 2ab[/tex]

Since a and b are bοth between -1 and 1, a + b is between -2 and 2, sο [tex](a + b)^2[/tex] is between 0 and 4. Therefοre, we have:

1 + 2ab ≤ 4

Simplifying, we get:

ab ≤ 3/2

This cοntradicts the fact that ab = 1, sο ab cannοt equal 1. Similarly, if ab = -1, we get:

[tex](a + b)^2 = 1 - 2ab[/tex]

Since [tex](a + b)^2[/tex] is nοnnegative, we have:

1 - 2ab ≥ 0

Simplifying, we get:

ab ≤ 1/2

This cοntradicts the fact that ab = -1, sο ab cannοt equal -1. Therefοre, we have -1 < ab < 1, which implies -1/2 ≤ ab ≤ 1/2.

Taking the square οf bοth sides οf -1/2 ≤ ab ≤ 1/2, we get:

[tex]1/4 \le a^2b^2 \le 1/4[/tex]

Adding [tex]a^2 + b^2 = 1[/tex]tο bοth sides, we get:

[tex]5/4 \le 1 + a^2b^2 \le 5/4[/tex]

Dividing by 2, we get:

[tex]5/8 \le (1 + a^2b^2)/2 \le 5/8[/tex]

Since [tex](1 + a^2b^2)/2[/tex] is the average οf [tex]a^2[/tex] and [tex]b^2[/tex], we have:

[tex]5/8 \le (a^2 + b^2)/2 \le 5/8[/tex]

Simplifying, we get:

5/8 ≤ 1/2 ≤ 5/8

Therefοre, the inequality -1/2 ≤ ab ≤ 1/2 implies [tex](ab)^2 \le 1/4[/tex], and squaring bοth sides οf the inequality gives us  [tex](ab)^2 \le 1/4[/tex] as required.

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If a rectangle is 10cm long and its perimeter is 26 cm, then the breadth of the rectangle is 3 cm

Let's assume the breadth of the rectangle to be 'b' cm.

We know that the perimeter of a rectangle is the sum of the lengths of all four sides.

Therefore, the perimeter of the given rectangle = 2(length + breadth)

Given, the length of the rectangle = 10 cm and the perimeter = 26 cm.

So, 2(10 + b) = 26

Simplifying this equation, we get:

20 + 2b = 26

Move 20 to right hand side of the equation

2b = 26 - 20

2b = 6

b = 6/2

b = 3 cm

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Answer:

The value of x is -2.

Step-by-step explanation:

(x+2) (x+3) - (x+2) (x-3) =0

Multiplying

x(x+3) +2(x+3) - [x(x-3) +2(x-3) =0

x² +3x +2x +6 -[x²-3x +2x -6] =0

Opening the bracket

x²+3x +2x +6 -x²+3x-2x+6=0

Adding the like terms

6x+12=0. [x²-x²=0]

6x = -12

x = -12 ÷6

x = -2

Answer:

A. 10/21

Step-by-step explanation:

There is a total of 7 marbles, 5 of which are red ---> 5/7

When you pull out a red marble and do not replace it in the bag before pulling the second one, a marble (red) is gone hence ----> 4/6

Multiply the two events to find the probability:

5/7 x 4/6 = 10/21

Janet will cover 416 square inches of rectangular cake with frosting.

What exactly is a rectangle?

A rectangle is a geometric shape that has four sides and four right angles (90 degrees). It is a type of quadrilateral, which means a four-sided polygon. The opposite sides of a rectangle are parallel and equal in length, and the adjacent sides are perpendicular to each other.

The area of a rectangle can be calculated by multiplying its length and width (or base and height) together, while its perimeter is the sum of the lengths of all four sides.

Now,

Since Janet is frosting the top layer of 4 rectangular birthday cakes, we need to calculate the total area of the top layer of all 4 cakes combined.

Each cake measures 13 inches by 8 inches, so the area of one cake is:

13 inches x 8 inches = 104 square inches

The top layer of one cake will have the same dimensions, so the area of the top layer of one cake is also 104 square inches.

To find the total area of the top layer of all 4 cakes combined, we can multiply the area of one cake by 4:

Total area = 104 square inches/cake x 4 cakes = 416 square inches

Therefore, Janet will cover 416 square inches of cake with frosting.

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Carlos' daughter will receive $847,438.00 from the savings account.

The formula used to calculate the amount of money earned through compound interest is [tex]A = P(1 + r/n)^nt[/tex], where A is the amount of money earned, P is the principal amount, r is the rate of interest per annum, n is the number of times the interest is compounded per year, and t is the number of years.

In Carlos' case, P = 400,000, r = 3.5%, n = 1, and t = 18 (the number of years between when his daughter was born and when she graduates high school). Plugging these values into the formula above, we get:

[tex]A = 400,000(1 + 0.035/1)^(1*18)A = 400,000(1.035)^18\\A = 400,000(2.11859)\\A = $847,438.00[/tex]

Therefore, upon graduation, Carlos' daughter will receive $847,438.00 from the savings account.

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There are 24 possible 3 digit numbers that can be formed using numerals 3,2,7 and 9 without repetition.

Since repetition is not allowed, the first digit can only be 3, 2, 7 or 9. Each of the four digits can be chosen in 4 ways.

For the second digit, the choices are now reduced to 3, 2 or 7 (since 9 has already been used). There can be 3 ways to pick the second digit.

For the third digit, the choices are now reduced to 2 or 7 (since 3 and 9 have already been used). There can be 2 ways to pick the third digit.

Therefore, there are 4 x 3 x 2 = 24 possible 3 digit numbers that can be formed using numerals 3,2,7 and 9 without repetition.

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Answer: Two parallel lines have the same slope. Therefore, we can use the slope of the given line y = -4x + 5 to find the slope of the line that is parallel to it.

The slope of y = -4x + 5 is -4. Therefore, the slope of any line parallel to it will also be -4.

Now we have the slope of the line and a point that it passes through. We can use point-slope form to write the equation of the line:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is the point on the line.

Plugging in the values we know, we get:

y - (-1) = -4(x - (-2))

y + 1 = -4(x + 2)

y + 1 = -4x - 8

y = -4x - 9

Therefore, the equation of the line that is parallel to y = -4x + 5 and passes through the point (-2,-1) is y = -4x - 9.

Step-by-step explanation:

The measure of angle abc is 64 degrees.

An angle bisector is simply a line or ray that divides an angle into two equal parts.

Given that;

Line bm bisects angle abc.

First, we determine the measure of angle abm.

Since line bm bisects angle abc, it divides the angle into two equal parts.

Hence;

Angle mbc = angle abm

Angle abc = angle mbc + abm

Plug in the value

Angle abc = 32 degrees + 32 degrees

Angle abc = 64 degrees.

Therefore, 64 degrees is the measure of ∠b.

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Answer:

on dividing both side of the second eqn by 6 you get

y+1/2x=2

so they are parallel to each other

[

keep in mind that:

any two equations are parallel if they have same coefficient for x and y but different constant values.

]

Answer:

slope(m1)= -coeff.of x/coeff.of y

:.m1= -1/2

slope(m2)= -coeff.of x/coeff.of y

:.m2= -1/2

so it is parallel to each other......

Answer:87.84$

Step-by-step explanation:

72------100%

x----------122%

x=122*72/100=87.84$

In step 4 and 5 he used prime factorization of 5 under cube root.

This was done in order to eliminate the cube root.

Prime factorization is a process of finding the prime factors of a given number. A prime factor is a factor of a number that is a prime number. The prime factorization of a number is the list of its prime factors, together with their multiplicities.

To find the prime factorization of a number, divide it by its smallest prime factor and write down the result as the product of the prime factor and the quotient. Continue this process with the quotient until quotient of 1 is obtained.  

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Answer:

B

Step-by-step explanation:

Permieter = 2(l + w)

200ft = 2(l + w)

l = 4 + 2w

200ft = 2(4  + 2w + w)

200ft = 2(4 + 3w)

200ft = 8 + 6w

6w = 192ft

w = 32ft

l = 4 + 2(32ft)

= 4 + 64ft

= 68ft

So the answer is B

Hope this helps!

Brainliest and a like is much appreciated!

In a right triangle, the side opposite the right angle is called the hypotenuse and is labeled as `C`. The other two sides are called the legs and are labeled as `A` and `B`. The angle opposite side `A` is labeled as `x`.

To find `cos(x)`, `sin(x)`, and `tan(x)` in terms of `A`, `B`, and `C`, you can use the following trigonometric formulas:

- `cos(x) = A/C`

- `sin(x) = B/C`

- `tan(x) = sin(x) / cos(x) = B/A`

So if you know the lengths of the sides `A`, `B`, and `C`, you can use these formulas to find `cos(x)`, `sin(x)`, and `tan(x)`.

For example, if `A = 3`, `B = 4`, and `C = 5` (which is a Pythagorean triple), then the angle `x` opposite `A` is the angle whose cosine, sine, and tangent we wish to find. Using the formulas above, we have:

- `cos(x) = A/C = 3/5`

- `sin(x) = B/C = 4/5`

- `tan(x) = sin(x) / cos(x) = (4/5) / (3/5) = 4/3`

Therefore, in this case, `cos(x) = 3/5`, `sin(x) = 4/5`, and `tan(x) = 4/3`.

Answer:

-16t^2 + 120t + 50

Step-by-step explanation:

Using the given values, we can write the model for this situation as:

h(t) = -16t^2 + 120t + 50

To determine how long it takes the thrown object to reach maximum height, we need to find the time at which the object reaches its maximum height. The maximum height occurs at the vertex of the parabolic path, which is given by:

t = -b / 2a

where a = -16, b = 120.

Substituting these values, we get:

t = -120 / 2(-16) = 3.75 seconds

Therefore, it takes 3.75 seconds for the thrown object to reach maximum height.