1. A rhombus shaped field has green grass for 18 cows to graze. If each side of the rhombus is 30 m and its longer diagonal is 48 m, how much area of grass field will each cow be getting?
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1. A rhombus shaped field has green grass for 18 cows to graze. If each side of the rhombus is 30 m and its longer diagonal is 48 m, how much area of grass field will each cow be getting?
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Answer:
Certainly! The area of a rhombus can be calculated using the formula A = [tex] \sf \frac{d_1 \times d_2}{2}[/tex], where d_1 and d_2 are the diagonals.
For the given rhombus:
- (d_1 = 30, m)
- (d_2 = 48, m)
Substitute these values into the formula:
A = [tex] \sf \frac{30 \times 48}{2}[/tex] = [tex] \sf \frac{1440}{2}[/tex] = 720 , [tex]{m}^2[/tex]
Now, to find the area each cow gets, divide the total area by the number of cows:
Area per cow = [tex]\frac{720}{m}^{2}\times{18}[/tex]
Calculating this gives:
Area per cow = 40, [tex]{m}^{2}[/tex]
Therefore, each cow will have 40 square meters of grass field to graze.
Answer:
[tex]\boxed{\bf\: Area \: of \: grass \: field \: each \: cow \: will \: graze = 48 \: {m}^{2} \: } \\ [/tex]
Step-by-step explanation:
Given that, each side of the rhombus is 30 m and its longer diagonal is 48 m.
Let assume that ABCD be a rhombus such that AB = BC = CD = DA = 30 m and diagonal AC = 48 m.
Now, Let's evaluate area of triangle ABC
So, we have
AB = 30 m
BC = 30 m
AC = 48 m
Now, Semi-perimeter (s) of triangle ABC is
[tex]\sf\: s = \dfrac{AB + BC + AC}{2} \\ [/tex]
[tex]\sf\: s = \dfrac{30 + 30 + 48}{2} \\ [/tex]
[tex]\sf\: s = \dfrac{108}{2} \\ [/tex]
[tex]\implies\sf\:s = 54 \: m \\ [/tex]
Now,
[tex]\sf\: Area \: of \: \triangle \: ABC \\ [/tex]
[tex]\sf\: = \: \sqrt{s(s - AB)(s - BC)(s - AC)} \\ [/tex]
[tex]\sf\: = \: \sqrt{54(54 - 30)(54 - 30)(54 - 48)} \\ [/tex]
[tex]\sf\: = \: \sqrt{54(24)(24)(6)} \\ [/tex]
[tex]\sf\: = \: \sqrt{(3)(3)(6)(24)(24)(6)} \\ [/tex]
[tex]\sf\: = \: 3 \times 6 \times 24 \\ [/tex]
[tex]\sf\: = \: 432 \: {m}^{2} \\ [/tex]
Thus,
[tex]\implies\sf\: Area \: of \: \triangle \: ABC = 432 \: {m}^{2} \\ [/tex]
Now,
[tex]\sf\: Area \: of \: rhombus \: ABCD = 2(Area\: of \: \triangle \: ABC) \\ [/tex]
[tex]\sf\: Area \: of \: rhombus \: ABCD = 2 \times 432 \\ [/tex]
[tex]\implies\sf\: Area \: of \: rhombus \: ABCD = 864 \: {m}^{2}\\ [/tex]
Now,
[tex]\sf\: Area \: of \: grass \: field \: each \: cow \: will \: graze = \dfrac{864}{18} = 48 \: {m}^{2} \\ [/tex]
Hence,
[tex]\implies\boxed{\bf\: Area \: of \: grass \: field \: each \: cow \: will \: graze = 48 \: {m}^{2} \: } \\ [/tex]