Question

The weekly number of tourists visiting an island is approximated by the equation given below y equals 30 plus 10 sine left bracket startfraction pi over 26 endfraction left parenthesis x minus 14 right parenthesis right bracket where y is the number of visitors​ (in thousands) in the xth week of the​ year, starting with x equals 1 for the first week in january. find the week in which the number of tourists to the island is

a. 40 comma 000​,
b. 35 comma 000​, and
c. 25 comma 000

Answer 1

1. When y = 40,000 x = 27th week.

2. When y = 35,000 x = 18th week.

3. When y = 25,000 x = 10th week.

Explanation:

Given:

y = 30 + 10sin[x/26(π-14)]

Required:

Find the value of x when

1. y = 40,000

2. y = 35,000

3. y = 25,000

To solve this we have to equate the expression with the value of y

1. y = 30 + 10sin[π/26 (x-14)]

When y = 40,(000)

We'll take y as 40.

So, we have

40 = 30 + 10sin[π/26 (x - 14)]

Collect like terms

40 - 30 = 10sin[π/26 (x - 14)]

10 = 10sin[π/26 (x - 14)]

Divide both sides by 10

10/10 = 10sin[π/26 (x-14)] ÷ 10

1 = sin[π/26 (x-14)]

Take sin inverse (arcsin) of both sides in radians

sin-¹(1) = [π/26 (x - 14)]

sin-¹(1) = ½π.. So, we have

½π = π/26 (x - 14)

Multiply both sides by 26/π

26/π * ½π = 26/π * π/26 (x - 14)

13 = x - 14

Make x the subject of formula

x = 13 + 14

x = 27.

Hence, when y = 40,000 x = 27th week.

2. y = 30 + 10sin[π/26 (x-14)]

When y = 35,(000)

We have

35 = 30 + 10sin[π/26 (x - 14)]

Collect like terms

35 - 30 = 10sin[π/26 (x - 14)]

5 = 10sin[π/26 (x - 14)]

Divide both sides by 10

5/10 = 10sin[π/26 (x-14)] ÷ 10

½ = sin[π/26 (x-14)]

Take sin inverse (arcsin) of both sides in radians

sin-¹(½) = [π/26 (x - 14)]

sin-¹(1) = π/6.. So, we have

π/6 = π/26 (x - 14)

Multiply both sides by 26/π

26/π * π/6 = 26/π * π/26 (x - 14)

26/6 = x - 14

4.3 = x - 14

Make x the subject of formula

x = 4.3 + 14

x = 18.3

x = 18 --- Approximated

Hence, when y = 35,000 x = 18th week.

3. y = 30 + 10sin[π/26 (x-14)]

When y = 25,(000)

So, we have

25 = 30 + 10sin[π/26 (x - 14)]

Collect like terms

25 - 30 = 10sin[π/26 (x - 14)]

-5 = 10sin[π/26 (x - 14)]

Divide both sides by 10

-5/10 = 10sin[π/26 (x-14)] ÷ 10

-½ = sin[π/26 (x-14)]

Take sin inverse (arcsin) of both sides in radians

sin-¹(-½) = [π/26 (x - 14)]

sin-¹(-½) = -π/6.. So, we have

-π/6 = π/26 (x - 14)

Multiply both sides by 26/π

-26/π * π/6= 26/π * π/26 (x - 14)

26/6 = x - 14

-4.3 = x - 14

Make x the subject of formula

x = -4.3 + 14

x = 9.7

x = 10 --- Approximated

Hence, when y = 25,000 x = 10th week.

Answer 2

The first thing we must do for this case is to rewrite the function correctly:
"y equals 30 plus 10 sine left bracket startfraction pi over 26 endfraction left parenthesis x minus 14 right parenthesis right bracket"
Rewriting:
y = 30 + 10 * sine [pi / 26 * (x-14)]
We substitute the value of y for each case and clear x.
We have then:
a. 40 comma 000:
40 = 30 + 10 * sine [pi / 26 * (x-14)]
x = 27

b. 35 comma 000
35 = 30 + 10 * sine [pi / 26 * (x-14)]
x = 18.33
c. 25 comma 000
25 = 30 + 10 * sine [pi / 26 * (x-14)]
x = 9.66