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A) Calculate the length of BC. b) Find the value of angle BAC in degrees, correct to 2 decimal places using tangent tables only. 40. Determine the value of x.

Mathematics

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39.

a) Calculate the length of BC

  1. Use the Pythagorean theorem for the right-angled triangle ABC.

    AC2=AB2+BC2AC^2 = AB^2 + BC^2

  2. Substitute the given values.

    (85 cm)2=(13 cm)2+BC2(85 \text{ cm})^2 = (13 \text{ cm})^2 + BC^2 7225 cm2=169 cm2+BC27225 \text{ cm}^2 = 169 \text{ cm}^2 + BC^2

  3. Solve for BC2BC^2.

    BC2=7225 cm2−169 cm2BC^2 = 7225 \text{ cm}^2 - 169 \text{ cm}^2 BC2=7056 cm2BC^2 = 7056 \text{ cm}^2

  4. Calculate BC.

    BC=7056 cm2BC = \sqrt{7056 \text{ cm}^2} BC=84 cmBC = 84 \text{ cm}

    The length of BC is 84 cm\boxed{\text{84 cm}}.

    b) The value of angle BAC in degrees, correct to 2 decimal places using tangent tables only.

  5. Identify the sides relative to angle BAC (θ\theta).

    Opposite side = BC = 84 cm
    Adjacent side = AB = 13 cm

  6. Use the tangent ratio.

    tan⁡(θ)=OppositeAdjacent=BCAB\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{BC}{AB} tan⁡(θ)=8413\tan(\theta) = \frac{84}{13} tan⁡(θ)≈6.461538\tan(\theta) \approx 6.461538

  7. Find the angle θ\theta.

    θ=arctan⁡(6.461538)\theta = \arctan(6.461538)

    Using a calculator (as an AI, I don't have physical tangent tables to interpolate from),

    θ≈81.2009∘\theta \approx 81.2009^\circ

  8. Round to 2 decimal places.

    θ≈81.20∘\theta \approx 81.20^\circ

    The value of angle BAC is 81.20∘\boxed{81.20^\circ}.

    40.
    Determine the value of xx.

  9. Use the trigonometric identity cos⁡(θ)=sin⁡(90∘−θ)\cos(\theta) = \sin(90^\circ - \theta).

    Given cos⁡(3x∘)=sin⁡((x−60)∘)\cos(3x^\circ) = \sin((x-60)^\circ).
    We can rewrite cos⁡(3x∘)\cos(3x^\circ) as sin⁡(90∘−3x∘)\sin(90^\circ - 3x^\circ).
    So, the equation becomes:

    sin⁡(90∘−3x∘)=sin⁡((x−60)∘)\sin(90^\circ - 3x^\circ) = \sin((x-60)^\circ)

  10. Equate the angles, since xx is an acute angle, 3x3x and x−60x-60 will be in ranges where this direct equality holds.

    90−3x=x−6090 - 3x = x - 60

  11. Solve for xx.

    90+60=x+3x90 + 60 = x + 3x 150=4x150 = 4x x=1504x = \frac{150}{4} x=37.5x = 37.5

    The value of xx is 37.5\boxed{37.5}.

    41.

    a) Calculate the amount of money in Ksh, that she obtained after exchanging the foreign currencies.

  12. Determine the exchange rate to use.

    Since the businesswoman arrived in Kenya with foreign currency, she is selling it to the bank. The bank will buy her currency.
    Buying rate for 1 Euro = 125 Ksh
    Buying rate for 1 Sterling Pound = 152 Ksh

  13. Calculate the amount from Sterling Pounds.

    1580 Sterling Pounds×152 Ksh/Sterling Pound=239960 Ksh1580 \text{ Sterling Pounds} \times 152 \text{ Ksh/Sterling Pound} = 239960 \text{ Ksh}

  14. Calculate the amount from Euros.

    1205 Euros×125 Ksh/Euro=150625 Ksh1205 \text{ Euros} \times 125 \text{ Ksh/Euro} = 150625 \text{ Ksh}

  15. Calculate the total amount obtained.

    239960 Ksh+150625 Ksh=390585 Ksh239960 \text{ Ksh} + 150625 \text{ Ksh} = 390585 \text{ Ksh}

    The businesswoman obtained 390585 Ksh\boxed{\text{390585 Ksh}}.

    b) Calculate the amount of money, in Ksh that she needed for the exchange.

  16. Determine the exchange rate to use.

    Since the trader is travelling abroad and needs foreign currency, she is buying it from the bank. The bank will sell her the currency.
    Selling rate for 1 Euro = 128 Ksh
    Selling rate for 1 Sterling Pound = 158 Ksh

  17. Calculate the cost for Euros.

    1250 Euros×128 Ksh/Euro=160000 Ksh1250 \text{ Euros} \times 128 \text{ Ksh/Euro} = 160000 \text{ Ksh}

  18. Calculate the cost for Sterling Pounds.

    1520 Sterling Pounds×158 Ksh/Sterling Pound=240160 Ksh1520 \text{ Sterling Pounds} \times 158 \text{ Ksh/Sterling Pound} = 240160 \text{ Ksh}

  19. Calculate the total amount needed.

    160000 Ksh+240160 Ksh=400160 Ksh160000 \text{ Ksh} + 240160 \text{ Ksh} = 400160 \text{ Ksh}

    The trader needed 400160 Ksh\boxed{\text{400160 Ksh}} for the exchange.

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