The diagram on the right shows the plan of a lake. The line segment $PQ$ represents the distance from the pier, $P$, to the far side of the lake. At equal intervals of 10 m along this line, perpendicular measures are made to the sides of the lake as shown. (a) (i) Use the Trapezoidal Rule to estimate the surface area of the lake. (ii) If the lake is on average 8 m deep, estimate the volume of water in the lake. (b) (i) Calculate the volume, $V$, in cm$^3$, of a hemisphere of radius 21 cm. Give your answer in terms of $\pi$. (ii) A buoy on the lake is in the shape of a hemisphere of radius 21 cm surmounted by a cone. Find $h$, the total height of the buoy.

Photo AI

The diagram on the right shows the plan of a lake - Leaving Cert Mathematics - Question 8 - 2021

Question 8

The diagram on the right shows the plan of a lake. The line segment $PQ$ represents the distance from the pier, $P$, to the far side of the lake. At equal intervals ... show full transcript

Worked Solution & Example Answer:The diagram on the right shows the plan of a lake - Leaving Cert Mathematics - Question 8 - 2021

Step 1

Use the Trapezoidal Rule to estimate the surface area of the lake.

96%

114 rated

Answer

To estimate the surface area, we need to identify the heights at the intervals along the segment $PQ$ . Based on the measurements given:

Heights at the intervals:
- At 0 m: 10 m
- At 10 m: 20 m
- At 20 m: 30 m
- At 30 m: 20 m
- At 40 m: 10 m

Using the Trapezoidal Rule:

$A = \frac{b-a}{2} (f(a) + f(b))$ where $b-a$ is the width, and $f(a)$ and $f(b)$ are the heights at the edges.

For this case, with an area breakdown:

Area parts:
- From 0 to 20 m: $A_1 = \frac{10}{2} (10 + 20) = 150$
- From 20 to 40 m: $A_2 = \frac{10}{2} (20 + 30) = 250$
- From 40 to 60 m: $A_3 = \frac{10}{2} (30 + 20) = 250$
- From 60 to 80 m: $A_4 = \frac{10}{2} (20 + 10) = 150$

Total area = $150 + 250 + 250 + 150 = 2050 \, m^2$ .

Step 2

If the lake is on average 8 m deep, estimate the volume of water in the lake.

99%

104 rated

Answer

To estimate the volume of the lake, we multiply the surface area by the average depth.

Using the previously calculated area:

$V = A \cdot d = 2050 \, m^2 \cdot 8 \, m = 16400 \, m^3$

Step 3

Calculate the volume, $V$, in cm$^3$, of a hemisphere of radius 21 cm. Give your answer in terms of $\pi$.

96%

101 rated

Answer

The formula for the volume of a hemisphere is given by:

$V = \frac{2}{3} \pi r^3$ Substituting $r = 21 \, cm$ :

$V = \frac{2}{3} \pi (21)^3 = 6174 \pi \, cm^3$

Step 4

Find $h$, the total height of the buoy.

98%

120 rated

Answer

The total height of the buoy consists of the height of the hemisphere and the height of the cone. The radius of the hemisphere is 21 cm, which contributes 21 cm to the height. The height of the cone can be calculated using the relationship with the radius:

Let the base of the cone be s, and it's height be h. From geometry, we find: $\frac{r}{h} = \frac{21}{s},$ where $s$ is given as 42 (from question context for the cone). Therefore:

$h = \frac{21^2}{42} = 21 \, cm$

Thus, total height: $Total \ Height = 21 \, cm (hemisphere) + 42 \, cm (cone) = 63 \, cm$

The diagram on the right shows the plan of a lake - Leaving Cert Mathematics - Question 8 - 2021

Worked Solution & Example Answer:The diagram on the right shows the plan of a lake - Leaving Cert Mathematics - Question 8 - 2021

Use the Trapezoidal Rule to estimate the surface area of the lake.

If the lake is on average 8 m deep, estimate the volume of water in the lake.

Calculate the volume, $V$, in cm$^3$, of a hemisphere of radius 21 cm. Give your answer in terms of $\pi$.

Find $h$, the total height of the buoy.

Join the Leaving Cert students using SimpleStudy...