section a 40 marks answer all the questions in this section. all questions carry equal marks. 1. an intercity shuttle travels between two cities, amanfro and kotoku, covering the first half of the distance at an average speed of 60 km/h and the second half at 90 km/h. the total distance between the two cities is 300 km. (a) calculate, for the entire journey, the average speed of the shuttle. (b) if the speed of the shuttle in the first half of the journey is reduced to 50 km/h, determine its effect on the overall time. 2. a stock - broker, kojo, invested some amount in two accounts which pay 11% and 10% simple interest annually. the amount invested at 11% is 140% of the amount invested at 10%. after 3 1/2 years, he earned a total of gh¢ 22,580.60 as interest. how much was invested at 11%? 3. a contractor builds a rainwater collection system in a shape of a hemispherical tank of radius 2 m to irrigate a 500 m² farm. each square metre of the farm requires 50 litres of water. determine, with a mathematical statement, if a full tank of water would be sufficient to irrigate the farm? take π = 22/7 4. from the top of a lighthouse 100 m high, the angles of depression of two fishing vessels are 48° and 36°. the foot of the lighthouse and the sea - level are on the same horizontal and the two vessels are on the same side of the lighthouse. (a) illustrate the information in a diagram. (b) find, correct to the nearest metre, the distance between the two vessels. 5. the mid - point of the line joining y(2a, 3) and z(-4, b) is m(1, 2a - 1). find: (a) the values of a and b; (b) |yz|.

Question

Accepted Answer

### 1. (a)
# Explanation:
## Step1: Calculate time for each half - distance
The total distance $d = 300$ km, so the half - distance $d_1=d_2 = 150$ km.
For the first half, speed $v_1 = 60$ km/h, time $t_1=\frac{d_1}{v_1}=\frac{150}{60}=2.5$ h.
For the second half, speed $v_2 = 90$ km/h, time $t_2=\frac{d_2}{v_2}=\frac{150}{90}=\frac{5}{3}$ h.
## Step2: Calculate total time and average speed
Total time $t=t_1 + t_2=2.5+\frac{5}{3}=\frac{5}{2}+\frac{5}{3}=\frac{15 + 10}{6}=\frac{25}{6}$ h.
Average speed $v=\frac{d}{t}=\frac{300}{\frac{25}{6}}=300	imes\frac{6}{25}=72$ km/h.
# Answer:
72 km/h

### 1. (b)
# Explanation:
## Step1: Calculate new time for first half - distance
The half - distance $d_1 = 150$ km, new speed $v_1'=50$ km/h, new time for first half $t_1'=\frac{d_1}{v_1'}=\frac{150}{50}=3$ h.
Original time for first half $t_1 = 2.5$ h (from part (a)).
## Step2: Calculate change in time
Change in time $\Delta t=t_1'-t_1=3 - 2.5 = 0.5$ h.
# Answer:
The overall time increases by 0.5 h.

### 2.
# Explanation:
## Step1: Let the amount invested at 10% be $x$.
Then the amount invested at 11% is $1.4x$.
The time $t = 3.5$ years.
Simple interest formula $I=Prt$, where $P$ is the principal, $r$ is the rate of interest and $t$ is the time.
Interest from 10% account $I_1=x	imes0.1	imes3.5 = 0.35x$.
Interest from 11% account $I_2=1.4x	imes0.11	imes3.5=1.4x	imes0.385 = 0.539x$.
## Step2: Calculate the total interest and solve for $x$
Total interest $I = I_1+I_2=0.35x + 0.539x=0.889x$.
Given $I = 22580.60$, so $0.889x=22580.60$.
$x=\frac{22580.60}{0.889}=25400$.
The amount invested at 11% is $1.4x$, so $1.4	imes25400 = 35560$.
# Answer:
GH₵ 35560

### 3.
# Explanation:
## Step1: Calculate the volume of the hemispherical tank
Volume of a hemisphere $V=\frac{2}{3}\pi r^{3}$, with $r = 2$ m and $\pi=\frac{22}{7}$.
$V=\frac{2}{3}	imes\frac{22}{7}	imes2^{3}=\frac{2}{3}	imes\frac{22}{7}	imes8=\frac{352}{21}\approx16.76$ m³.
Since $1$ m³ = 1000 litres, $V = 16760$ litres.
## Step2: Calculate the water required for the farm
The farm area is $A = 500$ m², and each m² requires 50 litres of water.
Water required $W=500	imes50 = 25000$ litres.
Since $16760<25000$, a full tank of water is not sufficient to irrigate the farm.
# Answer:
No

### 4. (a)
Draw a vertical line to represent the lighthouse of height $h = 100$ m. Mark two points on the same side of the base of the lighthouse to represent the two fishing vessels. Draw lines from the top of the lighthouse to the two points. Label the angles of depression as $48^{\circ}$ and $36^{\circ}$ respectively.

### 4. (b)
# Explanation:
## Step1: Let the distance of the nearer vessel from the base of the lighthouse be $x$ and the distance between the two vessels be $d$.
If the angle of depression of the nearer vessel is $48^{\circ}$, then $	an48^{\circ}=\frac{100}{x}$, so $x=\frac{100}{	an48^{\circ}}\approx100\div1.1106\approx90.04$ m.
If the angle of depression of the farther vessel is $36^{\circ}$, then $	an36^{\circ}=\frac{100}{x + d}$, so $x + d=\frac{100}{	an36^{\circ}}\approx100\div0.7265\approx137.65$ m.
## Step2: Calculate the distance between the two vessels
$d=\frac{100}{	an36^{\circ}}-\frac{100}{	an48^{\circ}}\approx137.65 - 90.04=47.61\approx48$ m.
# Answer:
48 m

### 5. (a)
# Explanation:
## Step1: Use the mid - point formula
The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.
For points $Y(2a,3)$ and $Z(-4,b)$ and mid - point $M(1,2a - 1)$:
$\frac{2a-4}{2}=1$ and $\frac{3 + b}{2}=2a - 1$.
From $\frac{2a-4}{2}=1$, we have $2a-4 = 2$, $2a=6$, $a = 3$.
## Step2: Substitute $a$ into the second equation to find $b$
Substitute $a = 3$ into $\frac{3 + b}{2}=2a - 1$.
$\frac{3 + b}{2}=2	imes3-1=5$.
$3 + b = 10$, $b = 7$.
# Answer:
$a = 3$, $b = 7$

### 5. (b)
# Explanation:
## Step1: Use the distance formula
The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}$.
For $Y(2a,3)$ and $Z(-4,b)$ with $a = 3$ and $b = 7$ (from part (a)), so $Y(6,3)$ and $Z(-4,7)$.
$|YZ|=\sqrt{(-4 - 6)^{2}+(7 - 3)^{2}}=\sqrt{(-10)^{2}+4^{2}}=\sqrt{100 + 16}=\sqrt{116}\approx10.77\approx10.8$.
# Answer:
$\sqrt{116}\approx10.8$

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QUESTION IMAGE

Question

Explanation:

1. (a)

Step1: Calculate time for each half - distance

Step2: Calculate total time and average speed

Step1: Calculate new time for first half - distance

Step2: Calculate change in time

Step1: Let the amount invested at 10% be $x$.

Step2: Calculate the total interest and solve for $x$

Answer:

1. (b)