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Maths Case study (4 Marks)

Heron's Formula [NEW] · Gujarat Board (GSEB) STD 9 (GSEB - English Medium). 2 questions.

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Case study (4 Marks)

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Question 14 Marks
To beautify parks in a city, city municipal corporation decided to make triangular flower beds in parks as shown in Fig. The dimensions of a triangular flower bed are $75 m \times 80 m \times 85 m$. Based on this information answer the following questions:
Image
(i) If each triangular flower bed is to be fenced with two parallel wires one below the other than the length of the wire used is
(a) 120 m$\quad$(b) 240 m$\quad$(c) 260 m$\quad$(d) 480 m
(ii) The area of a flower bed is
(a) $300 \sqrt{42} m^2$$\quad$(b) $300 \sqrt{21} m^2$$\quad$(c) $600 \sqrt{21} m^2$$\quad$(d) $400 \sqrt{21} m^2$
(iii) If each triangular bed is an equilateral triangle of side 60 m, then its area is
(a) $900 \sqrt{3} m^2$$\quad$(b) $600 \sqrt{3} m^2$$\quad$(c) $1200 \sqrt{3} m^2$$\quad$(d) $400 \sqrt{3} m^2$
(iv) The area of an isosceles triangle with base 'a' and equal sides 'b' is given by
(a) $\frac{a}{4} \sqrt{4 b^2-a^2}$$\quad$(b) $\frac{b}{4} \sqrt{4 a^2-b^2}$$\quad$(c) $\frac{a}{2} \sqrt{2 b^2-a^2}$$\quad$(d) $\frac{b}{2} \sqrt{4 a^2-b^2}$
(v) If each triangular bed is in the form of an isosceles triangle with base 60 m and equal sides of length 40 m each, then area of a flower bed is
(a) $150 \sqrt{7} m^2$ $\quad$(b) $75 \sqrt{7} m^2$ $\quad$(c) $300 \sqrt{7} m^2$ $\quad$(d) $200 \sqrt{7} m^2$
Answer
(i) (d): Length of wire $=2$ (Perimeter of the triangular field)
$=2(75+80+85) m=480 m$
(ii) (c): We have, $2 s=240 \Rightarrow s=120$
Let $\Delta$ be the area of a flower bed. Then,
$\Delta=\sqrt{s(s-a)(s-b)(s-c)}=\sqrt{120(120-75)(120-80)(120-85)}=600 \sqrt{21} m^2$
(iii) (a): Using: Area $=\frac{\sqrt{3}}{4}(\text { Side })^2$, we obtain
Area of a flower bed $=\frac{\sqrt{3}}{4} \times(60)^2 m^2=900 \sqrt{3} m^2$
(iv) (a)
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Question 24 Marks
In India, sunlight is available in abundance. Solar panels harness this energy and convert it into electric power. Mr. Krishan installed a triangular shaped solar panel of size 9 m, 9 m and 10 m on the roof of his house. Assuming that in the peak sunlight hours, 1 m2 pannel produces 190 watts of electricity per hour, answer the related questions given below (Efficiency of the pannel and other factors are not accountable).
Image
(i) Perimeter of the solar panel is:
(a) 14 m $\quad$(b) 18 m $\quad$(c) 35 m $\quad$(d) 28 m
(ii) Are in $m ^2$ of the solar panel can be calculated using the formula
(a) $\sqrt{s(s-a)(s-b)(s-c)}$
(b) $\sqrt{(s-a)(s-b)(s-c)}$
(c) $\sqrt{s(a-s)(b-s)(c-s)}$
(d)$\frac{1}{2} \sqrt{s(s-a)(s-b)(s-c)}$
where s is the semi-perimeter of the panel and a, b and care its sides.
(iii) Area of solar panel is:
(a) $\sqrt{140}$ sq. m $\quad$(b) $10 \sqrt{14} sq \cdot m$
(c) $14 \sqrt{10}$ sq. m $\quad$(d) $100 \sqrt{14}$ sq. m
(iv) The whole panel will produce: [Take $\sqrt{14}=3.74$ ]
(a) $190 \times 3.7$ watts/hour
(b) $190 \times 37.4$ watts/hour
(c) $190 \times 374$ watts/hour
(d) $190 \times 0.374$ watts/hour
Answer
(i) (d) (ii) (a) (iii) (b) (iv) (b)
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