Length of a rectangle is 7 cm more than the width. The perimeter is 78 cm. Find the rectangles dimensions

The length of the shortest ladder is 16.64 ft.

In triangle ABC, using Pythagoras theorem

[tex]AC^{2} = AB^{2} + BC^{2}[/tex]  OR

[tex]L^{2} = h^{2} + (x+4)^{2}[/tex]       .....(1)

We also have similar triangles if we consider:

△ABC and △DEC

Which gives us the proportional relationship:

AB:DE = BC:EC

or    h/8 = (4+x) / x

h = 8( 4+x) /x      

Substituting this last expression for h into Eq [1] we get:

[tex]L^{2} = \frac{(8(x+4))^{2} }{x^{2} } + (x+4)^{2}[/tex]

Our aim is to now minimize L wrt x, and this involves looking for critical points of the derivative, so differentiation wrt x we get:

[tex]2L\frac{dL}{dx} = 2( 8 + \frac{32}{x} ) (-\frac{32}{x^{2} } ) + 2(x+4)[/tex]

At a critical point the derivative is zero and so we have the equation:

         =  [tex]2 (x + 4) ( 1 - \frac{256}{x^{3} } ) = 0[/tex]

From here we can find the value x :

x + 4 = 0                                                            1 - [tex]\frac{256}{x^{3} }[/tex]  = 0

x = -4                                                                 x  = [tex]\sqrt[3]{256}[/tex]

We require x > 0 so we neglect the -ve solution and only take +ve solution

therefore the minimum height of the ladder can be given by putting the value of x in eqn 1.

[tex]L^{2} = ( 8 + \frac{32}{\sqrt[3]{256} } )^{2} + ( \sqrt[3]{256} +4) ^{2}[/tex]

L² = 277.14

L = 16.64 ft

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Answer:

g(10)=-33, x=3

Step-by-step explanation:

g(x)=-4x+7. --------(1)

a) g(10)=?

since we know function x = -4x+7 to find function g(10) we will just place 10 whenever we see x

g(10)=-4(10)+7

g(10)=-40+7

g(10)= -33

b.)find x if g(x)=-5. ---------(ii)

g(x)=-4x+7. we will place (I) in (ii)

g(x)=-5

-4x+7=-5

-4x=-5-7

-4x=-12

cancel the minus(-) sign in both sides

4x=12

x=12/4

x=3

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Answer:

After 8 minutes ; altitude = 22,400 feet

Step-by-step explanation:

s1 = 2800t

s2 = -3200t + 48 000

2800t = -3200t + 48 000

28t = -32t + 480

7t = -8t + 120

7t + 8t = 120

15t = 120

15/15 t = 120/15

t = 8 minutes

s = 8 * (2800)  =   22400 feet

Answer:

I believe it's 4.5 seconds

Answer:

Step-by-step explanation:

since x=y, you can either substitute for x or for y,

lets do x

-3x - x = 4

-4x=4

x=-1

Now we can just plug in

-3(-1) - y = 4

3-y = 4

y=-1

Read the left colum top to bottom then the right column top to bottom, hope this helps

She needs to complete 20 laps for a total distance of 0.5 kilometer.

1 kilometers = 1000 meters

0.5 kilometers = 500 meters

Length of swimming pool = 25 meters

That means, she covers 25 meters in 1 lap.

Laps to cover 500 meters = 500/25

                                            = 20

Hence, she needs to do 20 laps.

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The  line with a slope of –5 that passes through the points (7,–2) and (8,q) has q as -7.

The equation of a line can be described as follows:

y = mx + b

where

Therefore, the line passes through  (7, –2) and (8, q) with a slope of - 5.

Hence, let's find q.

y = - 5x + b

Let's find y-intercept

-2 = -5(7) + b

b = -2  + 35

b = 33

Therefore,

y = -5x + 33

Hence, let's find q.

y = -5(8) + 33

q = -40 + 33

q = -7

Therefore, q = - 7

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The distance from the top of the tree to the tip of the shadow is 12.8 ft  

According to the Pythagorean theorem, In a right-angled triangle, the sum of the squares of the side will equal the square of the hypotenuse.
Here hypotenuse is the longest side.

=> Hoptenuse² = side² + side²

Here we have

The length of the tree = 10 feet

Length of shadow = 8 feet

Here we need to find the distance from the top of the tree to the tip of the shadow  

If we join the top of the tree to the tip of the shadow we will observe a right-angled triangle as shown below

From the figure,

AC² = AB² + BC²    [ ∵ ABC is right angle triangle ]

=> AC² = 10² + 8²  

=> AC² = 100 + 64

=> AC² = 164

=> AC = 2√41

=> AC = 12.8 ft

Therefore,

The distance from the top of the tree to the tip of the shadow is 12.8 ft  

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Answer:

b) 12.8 feet

Step-by-step explanation:

Now we have to,

→ find the required distance.

Forming the equation,

→ x² = 10² + 8²

Then the distance will be,

→ x² = 10² + 8²

→ x² = 100 + 64

→ x² = 164

→ x = √164

→ [ x = 12.8 feet ]

Hence, distance is 12.8 ft.

Answer:

The equation that is true is equation two

The simple interest is given as PRT/100, thus the simple percent interest per year is 1%.

Simple interest is a way to figure out how much interest will be charged on a sum of money at a specific rate and for a specific duration of time.

Unlike compound interest, which adds the interest from the principal of prior years to determine the interest of the following year.

As per the given,

Investment = $70

Investment after one year is $70.70

Interset = 70.70 - 70 = $0.70

The interest is given as = PRT/100

0.70 = PR(1)/100

R = (0.70×100)/70

R = 1%

Hence, "The simple interest is given as PRT/100, thus the simple percent interest per year is 1%".

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Answer:

============

Since both points have same y-coordinate, the distance between the points is the difference of x-coordinates:

The slope of the line is 0.393.

Slope is the inclination of a line with respect to the horizontal is measured numerically. The ratio of the vertical to the horizontal distance between any two points on a line, ray, or line segment is known as its slope in analytic geometry ("slope equals rise over run").

Given,

A data set has an r- value of 0.793. If the standard deviation of the x-coordinates is 5.591, and the standard deviation of the y-coordinates is 2.772.

We'll use the following formula for slope if the r- value (or Pearson's coefficient) and the standard deviations of the x and y coordinates are provided:

m = 8ₓ/8ₐ

where m is the slope of the best-fit line.

8ₓ = Standard deviation of the x-coordinates is 5.591,

8ₐ =  the standard deviation of the y-coordinates is 2.772.

Slope is:

m = 8ₐ/8ₓ

m = (0.793)(2.772)/5.591

m = 0.393

The slope of the line is 0.393.

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They should produce 1091 units each day to yield a minimum cost.

What is the vertex formula?

When the graph crosses its symmetry axes, the vertex formula aids in determining the vertex coordinate of a parabola. The vertex point is often represented by (h, k).

We know that a parabola's conventional equation is y=ax²+bx+c. The vertex in this case should be near the base of the U-shaped curve if the coefficient of x² is positive. The vertex should be at the peak of the U-shaped curve if the coefficient of x² is negative.

We have,

C = 70,000 - 120x + 0.055x²

C = 0.055x² - 120x + 70,000

Hence, for this equation, the graph will be parabola opening upward.

For that, the vertex will be the minimum cost

x = (-b) / 2a

Here, a = 0.055 , b = -120

Now, find and put the values of x.

minimum cost (x) = ( -(-120) ) ÷ ( 2 × 0.055 )

                           = 120 ÷ 0.11

                           = 120 ÷ 0.11

                           = 1090.90  ≈ 1091 units

The minimum cost will be yield for 1091 units.

Therefore, They should produce 1091 units each day to yield a minimum cost.

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The common ratio, r = -0.75, 20th term = -18.9056, exact sum of the infinite sequence = 1750.

a.

common ratio r = a2/a1

= -3360/4480

= -0.75.

r = a3/a2

= 2520/-3360

= -0.75

b.

[tex]a_{n}[/tex] = a*r^(n-1)

here a = 4480 and r = -0.75.

= 4480(-0.75)^(20-1)

= 4480(-0.75)^19

= 4480*(-0.00422)

[tex]a_{20}[/tex] = -18.9056.

c.

[tex]s_{n}[/tex] = a*((1-r^n)/(1-r))

= 4480*((1--0.75^4)/(1--0.75))

= 4480*((1-0.31640625)/(1--0.75))

= 4480*(0.68359375/1.75)

= 4480*0.390625

= 1750

Therefore the common ratio, r = -0.75, 20th term = -18.9056, exact sum of the infinite sequence = 1750.

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Answer:

-3

Step-by-step explanation:

You can use  to find the slope of a line going through  and .

I like to line up the points and subtract vertically. Then put 2nd difference over 1st difference.

(6,h)

minus

(7,-10)

------------

-1     h-(-10)

This also said to be equal to -7:

Multiply -1 on both sides:

Subtract 10 on both sides:

[texh=-3[/tex]

So if h=-3 then the slope of the line going through (6,h) and (7,-10) is -7.

Let's test it:

(6,-3)

minus

(7,-10)

----------

-1  7

So the slope is 7/-1=-7. This is the intended goal.

The check is good.

The equation in the given question is answered in main body.

What is equation?

In arithmetic , an equation may be a formula that expresses the equality of 2 expressions, by connecting them with the sign =

Main body:

according to question :

A) equation for Saturday sale = 84c +157d = 2640

equation for Sunday sale = 67c +134d = 2211

when subtracted we will get, 17c +23d = 429

This equation shows the difference in sales of these 2 days in terms of ticket sold and collection.

B) According to second part solution are c= 9 , d= 12.

equation given : 17c+23d = 429

now substituting value of c and d.

⇒17(9) + 23(12)

⇒153+276

⇒429 = R.H.S

Hence equation satisfies.

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As corresponding angles ∠5 and ∠8 are equal. Hence line x is parallel to line z by converse of corresponding angles theorem.

What does converse of corresponding angles theorem states?

If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel.

How can we show two lines are parallel to each other?

We can show lines are parallel by finding a pair of corresponding angles that are congruent.

Proving if two lines are parallel then third line is parallel too.

According to the given data:

       Therefore, ∠5 ≅ ∠7 (i.e. corresponding angles)

       ∠5 = ∠7

       ∠6 ≅ ∠8 (Corresponding angles are equal)

        ∴ Measure of angles is also equal ⇒ ∠6 = ∠8

       ∠5 = ∠8 (transitive property of equality of angles which says if

                               ∠a = ∠b and ∠b = ∠c then ∠a = ∠c)

       ∠5 ≅ ∠8 (If measure of angles is same then they are congruent)

Therefore, line x || line z

(Converse of the Corresponding Angles Theorem).

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The partners are assigned in the form of (1, 15) (2, 14) (3, 13) (4, 12) (5, 11) (6, 10)  (18, 7) (17, 8), and (16, 9).

A square number, also known as a perfect square, is an integer that is the square root of another integer, or the result of multiplying another integer by itself.

For example:

Square of 2 = 4

Square of 3 = 9

Square of 4 = 16

Here we have  

A science class is pretty small

There are just 18 students in the class.

The teacher gave each of them numbers from 1 through 18 on lab days

Start from 1 and find by adding which number we will get a square number

Here  1 + 15 = 16

2 + 14 = 16

3 + 13 = 16

4 + 12 = 16

5 + 11 = 16

6 + 10 =16

After the above combinations, the remaining numbers are

7, 8, 9, 16, 17, 18          

which can be added as given below to get square number 25  

=>  18+7=25

=>  17+8=25

=>  16+9=25  

Therefore,

The partners are assigned in the form of (1, 15) (2, 14) (3, 13) (4, 12) (5, 11) (6, 10)  (18, 7) (17, 8), and (16, 9)  

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Answer:b=2/3 if u don't need an answer that is not a fraction or decimal u can write 17. as a whole number.

Step-by-step explanation:

The measure of the exterior angle as shown in the triangle is 140°.

Exterior angle is the angle between a side of a polygon and an extended adjacent side.

Exterior angle theorem: The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

From the diagram below, applying the theorem above,

Hence, the exterior angle is 140°.

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The annual dividend for 500 shares was $710.

The annual dividend is a function of the dividend per share and the number of shares held by a stockholder.

Dividends are always declared per share with a stated dollar amount or percentage.

Dividends represent the portion of the profits earned by an entity that it decides to distribute to the shareholders.

Dividends are a kind of financial reward to stockholders for their investment in the entity.

The dividend per share = $1.42

Market price per share = $96

The number of shares held = 500

The dividend received for the year = $710 ($1.42 x 500)

Thus, the holder of 500 shares in Walt Disney Co. will get $710 in a year the company declared and paid a $1.42 annual dividend.

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The x-intercept of the equation is 13 and its interpretation in the context of the problem is the number of time in hours for which Lily left her house.

In Mathematics, the x-intercept can be defined as the point at which the graph of a function crosses the x-coordinate (x-axis) and the value of "y" is equal to zero (0).

Next, we would determine the number of time for which Lily has left her house since in the morning by substituting the value of B with zero (0) as follows:

B = -4t + 52

0 = -4t + 52

4t = 52

Time, t = 52/4

Time, t = 13 hours.

Therefore, the x-intercept of the given equation is (0, 13).

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y = 0.1x + 1 is the linear function for pressure based on the depth in meters

Given that:  At 30 meters in depth the water pressure measures 4 (atm)

At 50 meters in depth under water the pressure is 6 (atm)

We will consider the two given data values as our points 1 and 2 respectively. And that is what we are going to use to create a linear function for pressure

Let the y and x represent the pressure and depth respectively

point 1 (30, 4) and point 2 (50, 6)

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

where  x1 =30, y1 = 4 and x2 = 50, y2 = 6

slope(m) = 6-4 / 50-30 = 2/20 = 1/10

Linear function are of the form y-y1 = m(x-x1). Thus:

y-4 = 1/10 (x-30)

y-4 = 1/10(x) - 3

y = 1/10(x) + 1 = 0.1x + 1

Therefore, the linear function for pressure based on the depth in meters is y = 0.1x + 1

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Answer:

50:75

Step-by-step explanation:

let the ratio of 2:3 be x

now,

125=2+3x

or,5x=125

therefore,x=125/5

                x=25

now,

2x=25*2

   =50

3x=25*3

    =75

The area of the triangle HKP found by the Heron's Formula is 136.23 sq. ft.

The three sides of the triangle are-

h = 24 ft, k = 21 ft, and p = 13 ft.

s = ( 24 + 21 + 13 ) / 2

s = 29

s - a = 29 - 24 = 5

s - b = 29 - 21 = 8

s - c = 29 - 13 = 16

Put in the formula,

Area = √s(s−a)(s−b)(s−c)

A = √ 29 x 5 x 8 x 16

A = 136.23

Thus, the area of the triangle HKP found by the Heron's Formula is 136.23 sq. ft.

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11.5 and 15 are the values that are part of the solution set of the inequality.

Inequality is defined as relationship between non-equal numbers or expressions. The solution of an inequality is the set of values that satisfies the given inequality.

To determine the solution set of the given inequality, isolate the variable to one side and simplify.

-4x + 24 < -2x + 2

-4x + 2x < -24 + 2

-2x < -22

22 < 2x

11 < x

Hence, the solution set of the given inequality is the set of numbers greater than 11. Among the given choices, 11.5 and 15 are both greater than 11.

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