A flag is posted on the top of a castle. A knight on the ground standing 30\,\text{m}30m30, start text, m, end text away from the castle notices that the angles of elevation of the top of the castle and the top of the flag are 30\degree30°30, degree and 45\degree45°45, degree respectively.
How long is the flag?
Answers
Answer:
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Answer:
Step-by-step explanation:
1 / 5
Strategy
Let's try to draw a rough diagram of the knight, the castle and the flag.
Here, \triangle ABD△ABDtriangle, A, B, D and \triangle ABC△ABCtriangle, A, B, C are right triangles.
We need to find the length of the flag, which is CDCDC, D or \purpleC{BD} - \maroonD{BC}BD−BCstart color #aa87ff, B, D, end color #aa87ff, minus, start color #ca337c, B, C, end color #ca337c.
Hint #22 / 5
Finding \maroonD{BC}BCstart color #ca337c, B, C, end color #ca337c
In \triangle BAC△BACtriangle, B, A, C, \tan(A)tan(A)tangent, left parenthesis, A, right parenthesis is the ratio of \maroonD{BC}BCstart color #ca337c, B, C, end color #ca337c and \blueD{AB}ABstart color #11accd, A, B, end color #11accd.
\begin{aligned} \tan(A)&= \dfrac{\maroonD{BC}}{\blueD{AB}}\\\\ \tan(30\degree) &= \dfrac{\maroonD{BC}}{\blueD{30}}\\\\ \dfrac{1}{\sqrt{3}} &= \dfrac{\maroonD{BC}}{30}\\\\ \maroonD{BC} &=\maroonD{10\sqrt{3}} \end{aligned}
tan(A)
tan(30°)
3
1
BC
=
AB
BC
=
30
BC
=
30
BC
=10
3
The castle is \maroonD{10\sqrt{3} \text{ m}}10
3
mstart color #ca337c, 10, square root of, 3, end square root, start text, space, m, end text, end color #ca337c high.
Hint #33 / 5
Finding \purpleC{BD}BDstart color #aa87ff, B, D, end color #aa87ff
In \triangle BAD△BADtriangle, B, A, D, \tan(A)tan(A)tangent, left parenthesis, A, right parenthesis is the ratio of \purpleC{BD}BDstart color #aa87ff, B, D, end color #aa87ff and \blueD{AB}ABstart color #11accd, A, B, end color #11accd.
\begin{aligned} \tan(A)&= \dfrac{\purpleC{BD}}{\blueD{AB}}\\\\ \tan(45\degree) &= \dfrac{\purpleC{BD}}{\blueD{30}}\\\\ 1 &= \dfrac{\purpleC{BD}}{30}\\\\ \purpleC{BD}&=\purpleC{30} \end{aligned}
tan(A)
tan(45°)
1
BD
=
AB
BD
=
30
BD
=
30
BD
=30
Castle ++plus Flag is \purpleC{30 \text{ m}}30 mstart color #aa87ff, 30, start text, space, m, end text, end color #aa87ff high.
Hint #44 / 5
Length of flag
We can subtract \maroonD{BC}BCstart color #ca337c, B, C, end color #ca337c from \purpleC{BD}BDstart color #aa87ff, B, D, end color #aa87ff to get the length of the flag.
\begin{aligned} \text{flag} &= \purpleC{BD} - \maroonD{BC}\\\\ &= \purpleC{30} - \maroonD{10\sqrt{3}} \end{aligned}
flag
=BD−BC
=30−10
3
Hint #55 / 5
The length of the flag is (30 - 10\sqrt{3}) \text{ m}(30−10
3
) mleft parenthesis, 30, minus, 10, square root of, 3, end square root, right parenthesis, start text, space, m, end text.