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Answer :
The direct distance from Tia to the goggles is approximately 2.38 feet.
Step 1
To find the direct distance from Tia to the goggles, we can use trigonometry, specifically the tangent function, which relates the angles of depression to the distances.
Let's denote:
- [tex]\( D_T \)[/tex] as the direct distance from Tia to the goggles.
- [tex]\( D_B \)[/tex] as the direct distance from Braden to the goggles.
- h as the depth of the pool (the vertical distance from the water surface to the goggles).
We know that Tia and Braden are 12 feet apart horizontally, and we need to find [tex]\( D_T \)[/tex].
Using the tangent function for the angles of depression:
[tex]\[ \tan(65^\circ) = \frac{h}{D_T} \][/tex]
[tex]\[ \tan(28^\circ) = \frac{h}{D_B} \][/tex]
We also know:
[tex]\[ D_T + D_B = 12 \][/tex]
Step 2
Let's express h in terms of [tex]\( D_T \)[/tex]and [tex]\( D_B \)[/tex] :
[tex]\[ h = D_T \tan(65^\circ) \][/tex]
[tex]\[ h = D_B \tan(28^\circ) \][/tex]
Equating the two expressions for h :
[tex]\[ D_T \tan(65^\circ) = D_B \tan(28^\circ) \][/tex]
Since [tex]\( D_B = 12 - D_T \)[/tex], substitute this into the equation:
[tex]\[ D_T \tan(65^\circ) = (12 - D_T) \tan(28^\circ) \][/tex]
Solving for [tex]\( D_T \)[/tex]:
[tex]\[ D_T \tan(65^\circ) = 12 \tan(28^\circ) - D_T \tan(28^\circ) \][/tex]
[tex]\[ D_T (\tan(65^\circ) + \tan(28^\circ)) = 12 \tan(28^\circ) \][/tex]
[tex]\[ D_T = \frac{12 \tan(28^\circ)}{\tan(65^\circ) + \tan(28^\circ)} \][/tex]
Step 3
Now we calculate the values:
[tex]\[ \tan(65^\circ) \approx 2.1445 \][/tex]
[tex]\[ \tan(28^\circ) \approx 0.5317 \][/tex]
Substituting these into the equation:
[tex]\[ D_T = \frac{12 \cdot 0.5317}{2.1445 + 0.5317} \][/tex]
[tex]\[ D_T = \frac{6.3804}{2.6762} \][/tex]
[tex]\[ D_T \approx 2.38 \text{ feet} \][/tex]
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Rewritten by : Barada
Assuming the goggles is 65 °and 28 °respectively. The direct distance from Tia to the goggles is 5.64 feet.
Direct distance
We would be making use of sine law to find the direct distnace
4=180°-65°-28°
4=87°
Let a represent the distance
Distance x=12/sin4=a/sin28°
a=12×sin28°/sin87°
Hence:
12/sin87°=a/sin28°
a=5.64 feet
Inconclusion the direct distance from Tia to the goggles is 5.64 feet.
Learn more about direct distance here:https://brainly.com/question/25292087
