how can you use functions to solve real world problems ??

Answer:

Therefore,

[tex]x=y= 4\sqrt{2}=5.6568\ units[/tex]

Step-by-step explanation:

Given:

Consider In Right Angle Triangle ABC

∠B = 90°

∠C = ∠A = 45°

AB = y

BC = x = adjacent side

AC = 8 = hypotenuse

To Find:

x = ?

y = ?

Solution:

In Right Angle Triangle ABC by Cosine Identity we have

[tex]\cos C = \dfrac{\textrm{side adjacent to angle C}}{Hypotenuse}\\[/tex]

substituting the above given values we get

[tex]\cos 45 = \dfrac{BC}{AC}=\dfrac{x}{8}[/tex]

[tex]\dfrac{1}{\sqrt{2} } =\dfrac{x}{8}\\\therefore x=\dfrac{8}{\sqrt{2} } \\Rationalizing\ we\ get\\\therefore x=\dfrac{8}{\sqrt{2}}\times \dfrac{\sqrt{2} }{\sqrt{2}}}\\\therefore x=4\sqrt{2}=4\times 1.4142=5.6568\ units[/tex]

As The triangle is 45 - 45 - 90

It is an Isosceles Right triangle

[tex]x=y[/tex]..... Isosceles Triangle property

[tex]\therefore y=4\sqrt{2}=4\times 1.4142=5.6568\ units[/tex]

Therefore,

[tex]x=y= 4\sqrt{2}=5.6568\ units[/tex]

Answer:

To work this problem out you would do it as if it were just a simple equation

7x-19<16 (you would first have to add 19 to both sides)

7x<35 (then you would divide by 7 to get the variable by itself)

x<5 ( you answer would then be 5)

Step-by-step explanation:

[tex] \frac{2}{3} x + 5 = 1[/tex]

[tex] \frac{2}{3} x = 1 - 5[/tex]

[tex] \frac{2}{3} x = ( - 4)[/tex]

[tex]x = ( - 4) \div \frac{2}{3} [/tex]

[tex]x = ( - 4) \times \frac{3}{2} [/tex]

[tex]x = \frac{ - 12}{ \: \: \: 2} [/tex]

[tex]x = ( - 6)[/tex]

[tex]∴ \frac{2}{3} \times ( - 6) + 5 = 1[/tex]

Answer:

The number of adult tickets sold is 894 and the number of kid tickets is 750.

Step-by-step explanation:

Given:

A Six Flags theme park charges $30 for adults and $15 for kids.

Total of 1,644 tickets were sold.

Total amount of tickets $11,250.

Now, to find the number of adult tickets and kid tickets.

Let the number of kid tickets be [tex]x.[/tex]

And the number of adult tickets be [tex]y.[/tex]

So, the total number of tickets:

[tex]x+y=1644.[/tex].....(1)

Solving the equation we get the value of [tex]x[/tex]:

[tex]x=1644-y.[/tex]

Now, the total amount of tickets of adult and kids:

[tex]15x+30y=11250.[/tex]

So, by putting the value of [tex]x[/tex] we get:

[tex]15(1644-y)+30y=11250[/tex]

[tex]24660-15y+30y=11250[/tex]

[tex]24660+15y=11250[/tex]

Subtracting both sides by 24660 we get:

[tex]15y=-13410[/tex]

Dividing both sides by -15 we get:

[tex]y=894[/tex]

Thus number of adult tickets = 894.

Now, putting the value of [tex]y[/tex] in equation (1):

[tex]x+894=1644[/tex]

On solving we get:

[tex]x=1644-894[/tex]

[tex]x=750.[/tex]

So. the number of kid tickets = 750.

Therefore, the number of adult tickets sold is 894 and the number of kid tickets is 750.

Answer: 750

Step-by-step explanation:

Answer:

Tuesday : Total = 30 : 72 = 5 : 12

Step-by-step explanation:

Monday: 42 miles

Tuesday: 42 - 12 = 30 miles

Monday + Tuesday: 42 + 30 = 72 miles

Tuesday : Total = 30 : 72 = 5 : 12

Answer:

A) x=2, 8

Step-by-step explanation:

abs(x-5)=3

x-5=3, x-5=-3

x=3+5=8,

x=-3+5=2

Final answer:

option A.  x=2, x=8

Explanation:

To solve the equation |x-5|=3, we need to consider both cases regarding the absolute value function. The absolute value of an expression is equal to 3 if the expression itself is 3 or -3. Therefore, we have two equations to solve:

Solving the first equation:

Solving the second equation:

Thus, the solutions for the equation |x-5|=3 are x=2 and x=8, which corresponds to option A.

Answer:

For x = 10,  the value of the given function f(x)= 4(x − 11) − 9 is -13.

Step-by-step explanation:

Here, the given expression is:

f(x) = 4(x − 11) − 9

Now, given: f(x)  = -13

⇒ -13  = 4(x − 11) − 9

Now, solving for the value of x, we get:

-13  = 4(x-11) -9

⇒ -13 = 4 x - 44 - 9

or, -13 + 44  + 9 =  4x

or, 4 x = 40

or, x = 40/4 = 10

or, x = 10

Hence, for x = 10,  the value of the given function f(x)= 4(x − 11) − 9 is -13.

Answer: 36

Step-by-step explanation:

The legs are given to be :

3x - 1 and -x + 27

The base = 5x + 1

One of the properties of an isosceles triangle is that two sides are equal , that is the two legs are equal. This means that

3x - 1 = - x + 27

3x + x = 27 + 1

4x = 28

x = 7

To calculate the length of the base, substitute x = 7 into the length of the base given , that is

Length of the base = 5(7) + 1

L = 35 + 1

L = 36

Answer:

y+1=-3/2(x-3)

Step-by-step explanation:

m=(y2-y1)/(x2-x1)

m=(5-(-1))/(-1-3)

m=(5+1)/-4

m=6/-4

simplify

m=-3/2

y-y1=m(x-x1)

y-(-1)=-3/2(x-3)

y+1=-3/2(x-3)

Answer:the branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences. The two main types are differential calculus and integral calculus.

Step-by-step explanation:

Final answer:

Calculus is the mathematics of change, encompassing derivatives and integrals, which are fundamental for solving rates of change and accumulation problems in various fields, including engineering and economics.

Explanation:

Calculus is a branch of mathematics that focuses on the study of change. It is divided into two major areas: differential calculus and integral calculus. Differential calculus deals with the concept of a derivative, which measures how a function changes as its input changes. This can be visualized as the slope of the tangent line to a curve at a given point. Integral calculus, on the other hand, involves the calculation of an integral, which can be thought of as the area under a curve or the accumulation of quantities. Together, these concepts allow us to solve complex problems involving rates of change and accumulation.

Calculus is essential in many fields such as engineering, physics, economics, and even biology. For example, in engineering, calculus is used to model and solve problems related to motion, forces, and energy.

It's important to note that students who do not take calculus in high school can still pursue engineering by taking calculus in their first year of college. The understanding of dimensions and units within calculus also applies to physical quantities, exemplifying the principle that dimensions obey the rules of algebra.

If a student is planning to major in economics, a strong recommendation is to learn at least a little calculus. This math competency greatly facilitates the understanding of advanced economics and helps speed up the learning process, especially in constructing and analyzing algebraic models used to describe economic relationships and functions.

Answer:

7<k+3+2

7<k+5

2<k

which is same as k>2

Answer:

The function through which given point passes is f(x) = 24 x - 12 .

Step-by-step explanation:

Given as :

The points are

[tex]x_1[/tex] , [tex]y_1[/tex] = 1 , 12

[tex]x_2[/tex] , [tex]y_2[/tex] = 2 , 36

now, slope of the line

Let The slope of line = m = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

Or, m = [tex]\frac{36 - 12}{2-1}[/tex]

Or, m = 24

So The slope of line = m = 24

Now, equation of line in point-slope form

y - [tex]y_1[/tex] = m × (x - [tex]x_1[/tex] )

Or, y - 12 = 24 × (x - 1 )

or, y - 12 = 24 x - 24

or, y = 24 x - 24 + 12

or, y = 24 x - 12

or , f(x) = 24 x - 12

So, The function through which given point passes is f(x) = 24 x - 12 . Answer

Answer:

$.56 per can; $.49 per can; 5 cans for $2.45

Step-by-step explanation:

If it were a dollar for two cans, it's pretty easy to figure out each can is 50 cents.  So you use the same idea.  if you have x cans for y dollars, if you divide both numbers by x you get the price of 1 can.

3 cans for 1.68 is 1 can for 1.68/3 = .56 so 56 cents

5 cans for 2.45 is 1 can for 2.45/5 = .49 so 49 cents.

You could use this trick dividing by the price and find how many cans you need to but to pay 1 dollar.

3 cans for 1.68 is 3/1.68 for 1 dollar or 1.786 cans for 1 dollar.  Doesn't make a lot of sense since you can't but part of a can, but I wanted to show you how you could use the logic for other things.

Answer:

a. [tex]0.1\ miles[/tex]

b Slope: [tex]\frac{1}{10}[/tex] or [tex]0.1[/tex]

Step-by-step explanation:

The equation of a line that passes through the origin is:

[tex]y=mx[/tex]

Where "m" is the slope of the line.

By definition, Direct variation equations have the following form:

[tex]y=kx[/tex]

Where "k" is the Constant of variation.

Then, you can conclude that:

[tex]m=k[/tex]

a. Let be "d" the distance in miles that the train travels per gallon.

You can observe in the graph that when the train travels 10 miles, it uses 100 gallons. Then, you can set up the following proportion:

[tex]\frac{10}{100}=\frac{x}{1}[/tex]

Solving for "x", you get:

[tex]x=\frac{1}{10}\\\\x=0.1[/tex]

Then, it travels 0.1 miles per gallon.

b. The slope is also [tex]\frac{1}{10}[/tex] or [tex]0.1[/tex]. To check this, you can subsitute the coordinates of the point (100,10) into [tex]y=mx[/tex] and solve for "m" in order to find its value.

This is:

[tex]10=m(100)\\\\m=\frac{1}{10}[/tex]

Answer:

[tex]y=mx[/tex]

Step-by-step explanation:

[tex]y=mx[/tex]

where, y is the total cost of the erasers,

m is the numbers of erasers,

x is the cost of each erasers.

Given:- [tex]m=19, y=3[/tex]

by putting these value in the above equation,

i.e. [tex]y=mx[/tex]

we get,

[tex]3=19x[/tex]

[tex]x=\frac{3}{19}[/tex]

[tex]x=0.157[/tex]

Now for 45 erasers substitute the value of [tex]x=0.157[/tex]

so, we get [tex]y=45\times 0.157[/tex]

[tex]y=7.065[/tex]

cost of 45 erasers are $7.065

Answer:

y = mx + b

Step-by-step explanation:

m is slope

b is y-intercept

Answer:

The answer is B. 6 and 8.

Answer:

B) 6 and 8

Step-by-step explanation:

[tex]3 \times 8 = 24[/tex]

[tex]4 \times 6 = 24[/tex]

[tex]2 \times 12 = 24[/tex]

Answer:

x= 1; y= 0; z= -2 and w= 3.

Step-by-step explanation:

Given that,

1) x + 2y - z = 3

2) 2x - y + z - w = -3

3) y + 2z - w = -7

4) x + 3y + 2z + 2w = 3 ​

now, from 1) z = x + 2y -3 →→(5)

from 2) w = 2x -y + z +3

         ⇒w = 2x -y + x + 2y -3 +3  (from (5))

            w = 3x + y →→(6)

Now substitute (5) and (6) in 3), we get

y +2(x + 2y -3) - (3x + y) = -7

⇒ 4y - x = -1 →→→(7)

Now substitute (5) and (6) in 4), we get

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

⇒ 9x +9y = 9

⇒ x + y =1 →→→(8)

⇒ x= 1-y , substituting this in (7) gives 5y -1 = -1

⇒ y = 0 and x = 1

substituting these values in

(5) and (6) gives, z = -2 and w = 3

⇒ x= 1; y= 0; z= -2 and w= 3.

Answer:

26.82

Step-by-step explanation:

Answer:

$26.82

Step-by-step explanation:

Product Cost: $18

Markup: 49%

Selling Price = (49% / 100% * $18) + $18 = $8.82 + $18 = $26.82

This gives you a price markup of $26.82.

this is my first answer and it could very well be wrong, sorry ;(

Answer:

x>-4

Step-by-step explanation:

13-2x > 21

-13 -13

-2x >8

(-2x)/(-2) >(8)/(-2)

x>-4

Answer:

The answer is 3024

Step-by-step explanation:

you do 9!/(9-4)! which comes out to 3024

Answer:

[tex]a=2s-b-c[/tex]

Step-by-step explanation:

Given formula:

[tex]s=\frac{a+b+c}{2}[/tex]

To solve for [tex]a[/tex]

In order to solve for [tex]a[/tex] we will try to isolate [tex]a[/tex] on one side of the equation.

Steps to isolate [tex]a[/tex].

1) Multiplying both sides by [tex]2[/tex] to remove fractions.

[tex]2\times s=2\times \frac{a+b+c}{2}[/tex]

[tex]2s=a+b+c[/tex]

2) Subtracting both sides by [tex]b[/tex]

[tex]2s-b=a+b-b+c[/tex]

[tex]2s-b=a+c[/tex]

3) Subtracting both sides by [tex]c[/tex]

[tex]2s-b-c=a+c-c[/tex]

[tex]2s-b-c=a[/tex]

Thus, we successfully isolated [tex]a[/tex] on one side. The formula for [tex]a[/tex] can be given as:

∴ [tex]a=2s-b-c[/tex]

Answer:

The sports shop has 21 part-time employees.

Step-by-step explanation:

Solution:

What is 25% of 84?

.Y is 25% of 84

Equation: Y = P% * X

Solving our equation for Y

Y = P% * X

Y = 25% * 84

Converting percent to decimal:

p = 25%/100 = 0.25

Y = 0.25 * 84

Y = 21

Or another quicker way....

50% is half of the whole, and 25 is half of 50.

84/2/2

equals 21 too, I don't recommend to use it as row form.

The sports shop has 21 part-time employees, found by calculating 25% of 84.

The question asks how many part-time employees a sports shop with 84 employees has if 25% of them work part-time. To find the answer, we need to calculate 25% of 84 employees. The calculation is done as follows:

First, convert the percentage to a decimal by dividing it by 100: 25% = 0.25.

Then, multiply the total number of employees by this decimal: 84  times 0.25 = 21.

Therefore, the sports shop has 21 part-time employees.

Answer:

13.31803792

Step-by-step explanation:

Law of Cosines

a= 8

b= 18

c= ?

[tex]c^2=a^2b^2-2ab*cos(c)[/tex]

[tex]c^2=8^2-18^2-2*8*18*cos(43)[/tex]

c = 64 + 324 - 2 * 8 * 18 * cos(43)

cos(43) = 0.7313537016

Order of Operations

64 + 324 - 2 * 8 * 18 * 0.7313537016 = 177.3701339

Square Root the answer

[tex]\sqrt{177.3701339}[/tex] = 13.31803792

The amount spent at the movie store is $ 10.5

Given that Davis bought some candy for $ 35

This was seven dollars less than four times what he bought at the movie store

Which means that $ 35 is equal to seven dollars less than four times what he bought at the movie store

Let "m" be the amount spent at the movie store

$ 35 = seven dollars less than four times what he bought at the movie store

35 = 4m - 7

35 + 7 = 4m

42 = 4m

Thus the amount spent at the movie store is $ 10.5

Answer:

34/9

Step-by-step explanation:

3 7/9 = 9*3+7 =34

34/9

Answer:34/9

Step-by-step explanation:

9*3 = 27

27+7= 34 then add the denominator

[tex]log_{x}(1/8)=-3/2[/tex]

This is equal to...

[tex]x^{-3/2}=1/8[/tex]

We want "x" so we need to isolate it using laws of exponents...

[tex]x^{(-1)(3/2)}=1/8[/tex]

[tex]x^{3/2}=8[/tex] because [tex]x^{-a}=1/x^{a}[/tex]

[tex]x=\sqrt[3]{8^{2}}[/tex] because [tex]x^{a/b}=\sqrt[b]{x^{a}}[/tex]

x = 4

answer: B

Answer:

49 cm²

Step-by-step explanation:

The volume (V) of a cube is calculated as

V = s³ ← s is the length of the side, thus

s³ = 343 ( take the cube root of both sides )

s = [tex]\sqrt[3]{343}[/tex] = 7, thus

area of base = s² = 7² = 49 cm²

Answer:

Substitute the slope  and the coordinates of point D in the equation of the line  y=mx+b and then solve for b in each equation

Step-by-step explanation:

we know that

The first step is calculate the slopes of these two lines. Remember that if two lines are parallel then the slopes are the same (m1=m2) and if two lines are perpendicular then the slopes is equal to m1*m2=-1

The second step is substitute the slope m2 and the coordinates of point D in the equation of the line in slope-intercept form y=mx+b and then solve for b in each equation

Answer:

40° ( angle 2 and 3 are alternating angles, therefore angle 2=3)