Answer to Question #160715 in Math for Smayya Ruganyira

Question

In a right-angled triangle the altitude to the hypotenuse is 8 cm high. The hypotenuse is 20 cm long. Find the lengths of the segements of the hypotenuse (Hint: let the length of the segment be x.)

In a right-angled triangle, the median to the hypotenuse is 6.4 cm. What is the length of the hypotenuse?

A bridge crosses a river at an angle of 60°. If the length of the bridge is 170 m, what is the width of the river?

If you deposit 6 500 FRW com-pounded monthly into an account paying 8% annual interest compounded monthly, how much money will be in the account after 7 years?

If you deposit 8 000 FRW into anaccount paying 7% annual interest compounded quarterly, how long will it take to have 12 400 FRW in the account?

A car hire company with 24 cars uses 2 940 litres of petrol in 5 days. How long would 4 116 litres of petrol last if the company had 28 cars and the consumption rate does not change?

A new car falls in value by 30% a year. After a year, it is worth 84 000 FRW. Find the price of the car when it was new.       

Answer 1

Let $x=$ the length of the first segement of the hypotenuse. Then the length of the second segement of the hypotenuse is $20-x.$

$$8^2=x(20-x), 0<x<20$$

$$x^2-20x+64=0$$

$$(x-4)(x-16)=0$$

$$x_1=4, x_2=16$$

The length of the first segement of the hypotenuse is 4 cm.

The length of the second segement of the hypotenuse is 16 cm.

In a right-angled triangle the middle of the hypotenuse is the center of the the circumscribed circle. Then hypotenuse is twice the median to the hypotenuse

$$2\cdot6.4=12.8$$

 The length of the hypotenuse is 12.8 cm.

If a bridge crosses a river at an angle of 60° with the river bank, then

$$\sin 60\degree=\dfrac{width}{length}$$

$$width=length\cdot\dfrac{\sqrt{3}}{2}$$

$$width=85\sqrt{3}\ m$$

$$FV=P(1+\dfrac{r}{n})^{nt}$$

$$FV=6500(1+\dfrac{0.08}{12})^{12(7)}=11358.24$$

11358.24 FRW will be in the account after 7 years.

$$FV=P(1+\dfrac{r}{n})^{nt}$$

$$12400=8000(1+\dfrac{0.07}{4})^{4(n)}$$

$$(1.0175)^{4(n)}=1.55$$

$$4n=\dfrac{\ln(1.55)}{\ln(1.0175)}$$

$$4n=25.2616$$

$$n=6.3154$$

It will take 6.3154 years to have 12 400 FRW in the account.

$$\dfrac{2940}{24\cdot5}=24.5$$

$$n=\dfrac{4116}{24.5\cdot28}=6$$

A car hire company with 28 cars uses 4 116 litres of petrol in 6 days.

$$\dfrac{84000}{1-0.3}=120000$$

The price of the car when it was new was 120 000 FRW.