The diagram shows a tetrahedron.
AD is perpendicular to both AB and AC.
AB = 10 cm.
AC = 8 cm.
AD = 5 cm.
Angle BAC = 90°
Calculate the size of angle BDC.
Give your answer correct to 1 decimal place.
Answers
=================================================
Work Shown:
For now focus on triangle ABC. Since this is a right triangle, we can use the pythagorean theorem to find the hypotenuse BC.
a^2 + b^2 = c^2
(AC)^2 + (AB)^2 = (BC)^2
8^2 + 10^2 = (BC)^2
(BC)^2 = 164
BC = sqrt(164)
BC = 12.806248 approximately
-------------
Move onto triangle ABD. Find the missing side BD through the pythagorean theorem
a^2 + b^2 = c^2
(AD)^2 + (AB)^2 = (BD)^2
5^2 + 10^2 = (BD)^2
125 = (BD)^2
(BD)^2 = 125
BD = sqrt(125)
BD = 11.1803399 which is approximate
---------------
Move onto triangle ADC. We'll use the pythagorean theorem again
a^2 + b^2 = c^2
(AC)^2 + (AD)^2 = (CD)^2
8^2 + 5^2 = (CD)^2
89 = (CD)^2
(CD)^2 = 89
CD = sqrt(89)
CD = 9.433981 also approximate
---------------
Focus on triangle BCD. We have the following side lengths
BC = 12.806248 BD = 11.1803399CD = 9.433981Let
side d be opposite angle D, so d = BC = 12.806248 side b be opposite angle B, so b = CD = 9.433981side c be opposite angle C, so c = BD = 11.1803399Note the use of uppercase letters for angles and lowercase letters for side lengths.
We can then use the law of cosines to find the angle we're after
d^2 = b^2 + c^2 - 2*b*c*cos(D)
12.806248^2=9.433981^2+11.1803399^2-2*9.433981*11.1803399*cos(D)
163.999987837504=213.999997787893-210.950228380284*cos(D)
-210.950228380284*cos(D) = -50.000009950389
cos(D) = (-50.000009950389)/(-210.950228380284)
cos(D) = 0.237022781792173
cos(D) = arccos(0.237022781792173)
D = 76.2891111084541
D = 76.3
The size of the angle ∠BDC of the tetrahedron can be determined by
cosine rule.
The size of angle ∠BDC is approximately 71.1°
Reasons:
A tetrahedron is a pyramid with triangular faces.
By Pythagoras's theorem, we have;
[tex]\overline{DC}[/tex]² = [tex]\mathbf{\overline{AD}}[/tex]² + [tex]\mathbf{\overline{AC}}[/tex]²
∴ [tex]\overline{DC}[/tex]² = 8² + 5² = 89
[tex]\overline{DC}[/tex]² = 89
[tex]\overline{BC}[/tex]² = [tex]\overline{AB}[/tex]² + [tex]\overline{AC}[/tex]²
∴ [tex]\overline{BC}[/tex]² = 10² + 8² = 164
[tex]\overline{BC}[/tex]² = 164
[tex]\overline{DB}[/tex]² = [tex]\mathbf{\overline{AD}}[/tex]² + [tex]\mathbf{\overline{AB}}[/tex]²
[tex]\overline{DB}[/tex]² = 5² + 10² = 150
[tex]\overline{DB}[/tex]² = 150
By cosine rule, we have;
[tex]\overline{BC}[/tex]² = [tex]\overline{DB}[/tex]² + [tex]\overline{DC}[/tex]² - 2 × [tex]\mathbf{\overline{DB}}[/tex] × [tex]\mathbf{\overline{DC}}[/tex] × cos(∠BDC)
Therefore;
164 = 150 + 89 - 2 × √(150) × √(89) × cos(∠BDC)
2 × √(150) × √(89) × cos(∠BDC) = 150 + 89 - 164 = 75
[tex]cos( \angle BDC) = \mathbf{\dfrac{75}{2 \times \sqrt{150} \times \sqrt{89} }}[/tex]
[tex]\angle BDC = arccos \left(\dfrac{75}{2 \times \sqrt{150} \times \sqrt{89} } \right) \approx \mathbf{71.1 ^{\circ}}[/tex]
∠BDC ≈ 71.1°
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Hey there!
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Given:
6 gallons of light green paint uses [tex]\dfrac{3}{8}[/tex] gallon of white paint.
Liam has [tex]\dfrac{9}{16}[/tex] gallon of white paint.
To find:
Whether Liam have enough white paint to make 8 gallons of light green paint.
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So, the white paint for 1 gallon of light green [tex]=\dfrac{\dfrac{3}{8}}{6}[/tex]
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[tex]=\dfrac{3}{48}[/tex]
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It means, 1 gallons of light green uses [tex]\dfrac{1}{16}[/tex] gallon of white paint.
White paint for 8 gallon of light green [tex]=8\times \dfrac{1}{16}[/tex]
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She has [tex]\dfrac{9}{16}[/tex] gallon of white paint, which is greater than [tex]=\dfrac{8}{16}[/tex] gallons.
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If Liam has 9/16 gallon, he has enough white paint to make 8 gallons of light green paint
Given that 6 gallons of light green paint use 3/8 gallon of white paint, this can be expressed as:
3/8 gallons of white paint = 6 gallons of light green
We are to determine if 9/16 gallons will be enough to make 8 gallons of light green paint. This can be expressed as
9/16 gallons of white paint = x
Divide both expressions
3/8 * 16/9 = 6/x
2/3 = 6/x
Cross multiply
2x = 3 * 6
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x = 18/2
x = 9
This shows that if Liam has 9/16 gallon, he has enough white paint to make 8 gallons of light green paint
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New: 17x^5 + 5x³ + 4x - 7
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Step-by-step explanation:
–5(–2) =
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explanation:
a negative times a negative is a positive and 2 times 5 in 10.
Answer:
pretty sure its 10
Step-by-step explanation:
First you wanna know 2 negatives equal a positive. So all you do is multiply the number 5×2=10