the sum of two numbers is 40. the sum of the smaller and 5 times the larger is 144. find the numbers.

Answer:

A. [tex]a_n=200+(n-1)5[/tex]

Step-by-step explanation:

Given,

The initial sprinting of Freya = 200 yards,

Also,  She gradually increases her distance by 5 yards per day,

So, in second day her sprinting = 200 + 5 = 205 yards,

In third day = 205 + 5 = 210 yards,

In fourth day = 210 + 5 = 215 yards,

....................,  so on,.....

Hence, the sequence that shows the given situation,

200, 205, 210, 215, .........

Which is an A.P.

That having first term, a = 200,

Common difference, d = 5,

Thus, the explicit formula for the given situation is,

[tex]a_n=a+(n-1)d[/tex]

[tex]\implies a_n=200+(n-1)5[/tex]

Option A is correct.

Answer:

Option A.

Step-by-step explanation:

The explicit formula of an AP is

[tex]a_n=a+(n-1)d[/tex]              .... (1)

where,

a is the first of the AP,

d is common difference.

It is given that Freya starts by sprinting 200 yards and she gradually increases her distance, adding 5 yards a day.

200, 205, 210,..., 305

Here,

First terms = 200

Common difference = 5

Substitute a=200 and d=5 in equation (1), to find the required explicit model

[tex]a_n=200+(n-1)5[/tex]

Therefore, the correct option is A.

Given that the population doubles every ten years, k may be found using the population growth equation dy/dt=ky by computing ln(2)/10, or roughly 0.0693 annually.

We begin by thinking about the solution to the differential equation dy/dt = ky, where the population doubles every ten years, in order to determine the value of k. This differential equation can be solved generally as y(t) = y(0)e^kt, where y(0) is the beginning population.

Since there are ten years between population doubling, we can write: y(10) = 2y(0) = y(0)e^10k.

The result of dividing both sides by y(0) is 2 = e^10k.

We get: ln(2) = 10k by taking the natural logarithm on both sides.

After calculating k, we have k = ln(2)/10 ≈ 0.0693.

Thus, k's value is around 0.0693 per year.

The angle's rate of change when the bottom of the ladder is 8ft from the wall is

[tex]\dfrac{-1.3}{6} rad/s[/tex]

It is given the Length of Ladder [tex](h)[/tex] is [tex]10ft.[/tex]. and the distance between the bottom of the ladder to the wall [tex](r)[/tex] is [tex]8ft.[/tex] as shown in the below figure.

By using the Pythagoras Theorem

[tex]b=\sqrt{10^{2}-8^{2} }\\=\sqrt{36} \\=6ft.[/tex]

and

[tex]cos(\theta)= r/h\\cos(\theta)=r/10[/tex]

Differentiating both sides with respect to [tex]'t'[/tex] by using the chain rule

[tex]-sin(\theta)\dfrac{\mathrm{d}\theta }{\mathrm{d} t}=\dfrac{1}{10} \dfrac{\mathrm{d}r }{\mathrm{d} t}\\\\\dfrac{\mathrm{d}\theta }{\mathrm{d} t}=\dfrac{1}{10} \dfrac{\mathrm{d}r }{\mathrm{d} t} \dfrac{1}{-sin(\theta)} } ......(eq. 1)[/tex]

given

[tex]\dfrac{\mathrm{d}r }{\mathrm{d} t}=1.3ft./s\\sin(\theta)=\frac{6}{10}[/tex]

putting this in eq.1, we get

[tex]\dfrac{\mathrm{d} \theta}{\mathrm{d} t} = \dfrac{1.3}{10} (\dfrac{1}{-\frac{6}{10} }) \\\dfrac{\mathrm{d} \theta}{\mathrm{d} t} =\dfrac{-1.3}{6} rad/s[/tex]

So the angle's rate of change when the bottom of the ladder is 8ft from the wall is[tex]\dfrac{-1.3}{6} rad/s[/tex].

Know more about Pythagoras Theorem here:

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Answer:

Solution of the system of the equations is (0.882, 0.647)

Step-by-step explanation:

As shown in the figure equations are [tex]y=-\frac{2}{5}x+1[/tex] and y = 3x - 2

Solution of the system of equations will be the common point or point of intersection of these lines.

To get the point of intersection we will solve these equations.

We will equate these equations to get the value of x.

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

[tex]-2x+5=15x-10[/tex]

15x + 2x = 10 + 5

17x = 15

[tex]x=\frac{15}{17}[/tex]

x = 0.882

By putting x = 0.882 in y = 3x - 2

y = 3(0.882) - 2

y = 0.647

So the solution of systems of equations is (0.882, 0.647)

Answer:

The height of the ball after t =1.5 second is 7.5 feet.

Description:

The basket ball shoots from a height of 6 feet, it increase the height 1.5 feet after 1.5 seconds.

Therefore, the basketball thrown at the rate of 1 feet/per second.

Step-by-step explanation:

Given: A player shoots a basketball from a height of 6 feet. The equation,

h(t) = [tex]-16t^2 + 25t + 6[/tex]

We need to find the height of the ball when t = 1.5 and describe the height.

plug in t = 1.5 in h(t) = [tex]-16t^2 + 25t + 6[/tex]

h(1.5) = [tex]-16(1.5)^2 + 25(1.5) + 6[/tex]

Simplifying the above expression, we get

h(1.5) = -36 + 37.5 + 6

h(1.5) = -36+43.5

h(1.5) = 7.5 feet

The height of the ball after t =1.5 second is 7.5 feet

Description:

The basket ball shoots from a height of 6 feet, it increase the height 1.5 feet after 1.5 seconds.

Therefore, the basketball thrown at the rate of 1 feet/per second.

Answer:

B and C

Step-by-step explanation:

Answer:

y = $1,000/car x + $40,000

20 cars

Step-by-step explanation:

Let

x = number of cars sold

y = total earnings

The relationship between y and x can be expressed through a linear equation.

y = mx + b

where,

m is the slope

b is the y-intercept

The slope is how much she earns per car sold, that is, the commission of $1,000/car (I think you meant this number instead of $1,00).

The y-intercept is what she earns even if she does not sell any car, i.e. a salary of $40,000.

The resulting equation is

y = $1,000/car x + $40,000

If she is to make $60,000 in one year (y = $60,000), the number of cars sold is:

$60,000 = $1,000/car x + $40,000

$20,000 = $1,000/car x

x = 20 car

Answer:

  See the attachment

Step-by-step explanation:

You want a graph of the parabola defined by f(x) = x² +10x +24.

We can add and subtract the square of half the x-coefficient to put the equation into vertex form:

  f(x) = x² +10x +24 . . . . . . . . . . given

  f(x) = x² +10x +25 +24 -25 . . . . add and subtract (10/2)² = 25

  f(x) = (x +5)² -1 . . . . . . . . . write in vertex form

Comparing to the vertex form equation f(x) = (x -h)² +k, we see that ...

  (h, k) = (-5, -1).

These are the coordinates of the vertex.

The coefficient of x² is 1, so another point can be found by adding (1, 1) to the vertex. That means a second point on the parabola is ...

  (-5, -1) +(1, 1) = (-4, 0)

The attached graph shows the parabola with vertex (-5, -1) through point (-4, 0).

__

Additional comment

When the coefficient of x² is not 1, the process is a little different. If you start with f(x) = ax² +bx +c, you can factor 'a' from the first two terms to help you get vertex form.

  f(x) = a(x² +(b/a)x) +c
  f(x) = a(x² +(b/a)x +(b/(2a))²) + c - a(b/(2a))²
  f(x) = a(x +b/(2a)x)² +(c -b²/(4a))

The vertex is (-b/(2a), c -b²/(4a)).

The second point can be found by adding (1, a) to the vertex coordinates.

Answer:

Step-by-step explanation:

The answer is 25+26+27+28+29=135

Is the correct answer I basically divided 135 by 5 and got 27 then I just worked around that number to get the answer.

Answer:

An equivalent expression is [tex]6 - 16y[/tex]

Step-by-step explanation:

The distributive property consists in removing the parenthesis. For this problem, it is done by multiplying the number outside the parenthesis by each term inside the parenthesis and then performing a subtraction between the results because of the minus sign inside.

For the expression [tex]2(3-8y)[/tex], when applying the distributive property, you would get:

[tex]2\times 3 - 2\times 8y[/tex]

[tex]6 - 16y[/tex]

Thus, an equivalent expression is [tex]6 - 16y[/tex].

The distributive property to remove the parentheses in the expression 2(3 - 8y) is 6 - 16y

From the question, we have the following parameters that can be used in our computation:

2(3 - 8y)

When the expression is expanded, we have the following

2(3 - 8y) = 2 * 3 - 2 * 8y

Evaluate the products

This gives

2(3 - 8y) = 6 - 16y

Lastly, we have

2(3 - 8y) = 6 - 16y

Hence the expression when simplified is 6 - 16y

Read more about expression at

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