Case study (4 Marks)

Question 14 Marks

Rinku was very happy to receive a fancy jumbo pencil from his best friend Rohan on his birthday. Pencil is a basic writing tool, when sharpened its shape is a combination of cylinder & cone as given in the picture.
Cylindrical pencil with conical head is a common shape worldwide since ages. Commonly pencils are made up of wood & plastic but we should promote pencils made up of eco-friendly material (many options available in the market these days) to save environment.

The dimensions of Rinku’s pencil are given as follows:
Length of cylindrical portion is 21cm. Diameter of the base is 1 cm and height of the conical portion is 1.2 cm
Based on the above information, answer the following questions:
(i) Find the slant height of the sharpened part.
(ii) Find curved surface area of sharpened part (in terms of $\pi$).
(iii)(A) Find the total surface area of the pencil (in terms of $\pi$).
OR
(B)The pencil’s total height decreases by 8.2 cm after sharpening it many times, what is the volume of the cylindrical part of the shortened pencil (in terms of $\pi$)?

Answer

(i) $l^2=(1.2)^2+(0.5)^2$
= 1.44 + 0.25
$\Rightarrow l=\sqrt{1.69}=1.3 cm$
(ii) Curved surface area of sharpened part
$=\pi \times 0.5 \times 1.3$
$=(0.65 \pi) cm ^2$
(iii) (A) Total surface area of pencil
= CSA of cylinder + CSA of cone + area of base circle
$=\pi \times 0.5 \times 0.5 \times 21+0.65 \pi+\pi \times(0.5)^2$
= (5.25 + 0.65 + 0.25) $\pi$
$=(6.15 \pi) cm ^2$
OR
(B) Length of cylindrical part of shortened pencil
$=(21-8.2) cm =12.8 cm$
So, volume of cylindrical part of shortened pencil
$=\pi \times 0.5 \times 0.5 \times 12.8$
$=(3.2 \pi) cm ^3$

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Question 24 Marks

Radha, an aspiring landscape designer, is tasked with creating a visually captivating pool design that incorporates a unique arrangement of fountains. The challenge entails arranging the fountains in such a way that when water is thrown upwards, it forms the shape of a parabola. The graph of one such parabola is given below.

The height of each fountain rod above water level is 10 cm. The equation of the downward-facing parabola representing the water fountain is given by
$p(x)=-x^2+5 x-4$
Based on the above information, answer the following questions:
(i) Find the zeroes of the polynomial p(x) from the graph.
(ii) Find the value of x at which water attains maximum height.
(iii)(A) If h is the maximum height attained by the water stream from the water level of the pool, then find the value of h.
OR
(B) At what point(s) on x- axis, the height of water above x- axis is 2 m?

Answer

(i) 1 and 4
(ii) 𝑥 = $5 / 2$
(iii) (A) At $x=5 / 2, p(x)=2.25$
Therefore, $h=0.10+2.25=2.35 m$
OR
(B) $-x^2+5 x-4=2$
$x^2-5 x+6=0$
$(x-2)(x-3)=0$
$\Rightarrow x=2$ and $x=3$
Therefore, required points are (2,0) and (3,0)

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Question 34 Marks

A group of students conducted a survey to find out about the preferred mode of transportation to school among their classmates. They surveyed 200 students from their school. The results of the survey are as follows:
120 students preferred to walk to school.
25% of the students preferred to use bicycles.
10% of the students preferred to take the bus.
Remaining students preferred to be dropped off by car.
Based on the above information, answer the following questions:
(i) What is the probability that a randomly selected student does not prefer to walk to school?
(ii) Find the probability of a randomly selected student who prefers to walk or use a bicycle.
(iii)(A) One day 50% of walking students decided to come by bicycle. What is the probability that a randomly selected student comes to school using a bicycle on that day?
OR
(B) What is the probability that a randomly selected student prefers to be dropped off by car?

Answer

(i) Number of students who do not prefer to walk = 200 − 120 = 80
P (selected student doesn’t prefer to walk) = $\frac{80}{200}$ or $\frac{2}{5}$
(ii) Total number of students who prefer to walk or use bicycle = 120 + 50
= 170
P (selected student prefers to walk or use bicycle) = $\frac{170}{200}$ or $\frac{17}{20}$
(iii) (A) 50% of walking students who used bicycle = 60
Number of students who already use bicycle = 50
P (selected student uses bicycle) = $\frac{110}{200}$ or $\frac{11}{20}$
OR
(B) Number of students who preferred to be dropped by car
= 200 − (120 + 50 + 20)
= 10 students
P (selected student is dropped by car) = $\frac{10}{200}$ or $\frac{1}{20}$

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Maths Case study (4 Marks)

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