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3 years ago

Given the recursive formula for an arithmetic sequence find the explicit formula.

Mathematics

1 answer:

trasher [3.6K]3 years ago

3 0

A recursive formula is a function of the previous term. The "an-1" is the term that comes before "an".
an arithmetic sequence just adds or subtracts the same number (common difference, d) between terms. looking at the recursive formula it is -9
-35, -44, -53, -62, .....

The explicit formula is a function of the terms place "n" in the sequence. This is written as :
an = a1 + d(n-1)
fill in
a1 = -35 and d = -9

an = -35 - 9(n-1)
expand to
an = -35 - 9n + 9
an = -26 - 9n

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What is the missing angle in the right triangle?

Ainat [17]

Answer:

Step-by-step explanation:

For angle A, 3 is the adjacent leg. 8 is the hypotenuse.

The trig ratio that relates the adjacent leg to the hypotenuse is the cosine.

$\cos A = \dfrac{adj}{hyp}$

$\cos A = \dfrac{3}{8}$

$\cos A = 0.375$

We know the cosine of angle A equals 0.375. We take the inverse cosine of 0.375 to find the measure of the angle.

$A = \cos^{-1} 0.375$

$A = 68^\circ$

8 0

3 years ago

Find the points on the cone z2 = x2 + y2 that are closest to the point (6, 2, 0).

aliya0001 [1]

Answer:

The closest points on the cone are;

(6, 2, -√10) and (6, 2, √10)

Step-by-step explanation:

Let B(x, y, z) denote a point on the cone.

Therefore, the distance between the points (6, 2, 0) and B(x, y, z) is;

d = √[(x - 6)² + (y - 2)² + (z - 0)²]

d = √[(x - 6)² + (y - 2)² + z²]

Since we are given that z² = x² + y², we now have;

d = √[(x - 6)² + (y - 2)² + x² + y²]

Taking the square of both sides gives;

d² = [(x - 6)² + (y - 2)² + x² + y²]

x² is an increasing function. Thus, minimizing d is also the same as to minimize f (x, y) = d²

Thus, f' = 0. So;

df/dx = 2(x - 6) + 2x = 0

2x - 12 + 2x = 0

4x = 12

x = 12/4

x = 3

Similarly,

df/dy = 2(y - 2) + 2y = 0

2y - 4 + 2y = 0

4y - 4 = 0

4y = 4

y = 4/4

y = 1

Now,from earlier;

z² = x² + y²

Thus;

z = ±√(3² + 1²)

z = ±√10

Thus, the closest points on the cone are;

(6, 2, -√10) and (6, 2, √10)

7 0

4 years ago

Item 4 In a circle with a radius of 36.9 m, an arc is intercepted by a central angle of 8π 5 radians. What is the arc length? Us

leva [86]

Answer:

The arc length is 185.39 meters.

Step-by-step explanation:

The arc length is calculated by the following expression:

$\Delta s = \Delta \theta \cdot r$

Where:

$r$ - Radius, measured in meters.

$\Delta \theta$ - Central angle, measured in radians.

If $r = 36.9\,m$ and $\Delta \theta =\frac{8}{5}\pi\, rad$ , the arc length, measured in meters, is:

$\Delta s = \frac{8}{5}\pi\cdot (36.9\,m)$

$\Delta s = \frac{8}{5}\cdot (3.14)\cdot (36.9\,m)$

$\Delta s \approx 185.386\,m$

$\Delta s \approx 185.39\,m$

The arc length is 185.39 meters.

5 0

3 years ago

Need Help

klio [65]

The smallest amount of material needed is 54 square centimeters

<h3>How to determine the amount of material needed?</h3>

The given parameters are:

Volume = 36 cubic centimeters

Represent length with x, width with y and height with z.

So, we have

x = 3y

The volume is calculated as:

V = xyz

This gives

V = 3y²z

Substitute 36 for V

3y²z = 36

Divide by 3

y²z = 12

Make z the subject

z = 12/y²

The surface area is:

S = 2(xy + xz + yz)

This gives

S = 2(3y² + 3yz + yz)

Evaluate the like terms

S = 2(3y² + 4yz)

Expand

S = 6y² + 8yz

Substitute z = 12/y²

S = 6y² + 8y * 12/y²

This gives

S = 6y² + 96/y

Differentiate

S' = 12y - 96/y²

Set to 0

12y - 96/y² = 0

Multiply through by y²

12y³ - 96 = 0

Add 96 to both sides

12y³ = 96

Divide by 12

y³ = 8

Take the cube root of both sides

y = 2

Recall that:

x = 3y and z = 12/y²

This gives

x = 3 * 2 = 6

z = 12/2² = 3

Recall that:

S = 2(xy + xz + yz)

So, we have:

S = 2(6 * 2 + 3 * 3 + 2 * 3)

Evaluate

S = 54

Hence, the smallest amount of material needed is 54 square centimeters