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°
Learn more here:
https://brainly.com/question/20839703
Related Questions
A video game which originally sold for $40 was reduced $5. Find the
percent of reduction.
Answers
Answer:
12.5%
Step-by-step explanation:
Solve each equation by using the square root property. 4X^2+9+=33
Answers
Answer:
Step-by-step explanation:
4X^2=33-9
4X^2=24
X^2=24:4
X^2=6
X=\sqrt{6}
Good luck!
Julisa had a coupon for 15% discount off the regular price of a pair of shoes at the stores. The regular price of the shoes she bought was $38.95. What was the discounted price of the shoes not including tax?
Answers
Answer:
33.10
Step-by-step explanation:
how do you make a fraction that’s greater than one or just a fraction into a mixed number?
Answers
Answer:
You multiply the denominator and the whole number then add the numerator.
Step-by-step explanation:
Example: [tex]3\frac{2}{4}[/tex]
You multiply 3 and 4
3*4=12
Then add the numerator
12+2=14
The final answer would be
[tex]\frac{14}{4}[/tex]
Y = 2x/a + 2b solve for b
Answers
Answer:
b = Y/2 - x/a
Step-by-step explanation:
HELP ME PLEASE!!!!!
this is hard for me i need help
Answers
Answer:
-1 1/2, 3
Step-by-step explanation:
A cone has surface area 360π in^2 and volume 800π in^3. the cone is dilated, and the surface area of the dilated cone is 2,250π in^2. what is the dilated cone's volume? asap! please my mom is trying to hurt me im locked in my closet! my dads home!!!!!!!!
they're trying to make me eat something called mellotonin while my mom is sniffing sugar in the livingroom!!!!
Answers
The dilated cone's with surface area of 2,250π in^2 has a volume of about 12500π in³
What is dilation?
Dilation is the increase or decrease in the size of a figure.
The cone has surface area 360π in² and volume 800π in^3. the cone is dilated, and the surface area of the dilated cone is 2,250π in^2.
Hence:
Scale factor² = dilated cone area / cone surface area
scale factor² = 2250π / 360π
Scale factor = 2.5
Volume of dilated cone = 2.5³ × 800π = 12500π in³
The dilated cone's with surface area of 2,250π in^2 has a volume of about 12500π in³
Find out more on dilation at: https://brainly.com/question/10253650
Hich would be a good statistical question Alexandra could have used to create the tally chart? How many people read three books in the last month? How many people read one or two books in the last month? Did you read in the last month? How many books did you read in the last month?
Answers
Incomplete questions. Here's the complete question:
Alexandra made a tally chart shown below. A 2-column table with 5 rows. Column 1 is labeled Books Read in the Last Month with entries 1, 2, 3, 4, 5. Column 2 is labeled Tally with entries 2, 5, 3, 1, 2.
Which would be a good statistical question based on the tally chart?
How many people read three books in the last month?How many people read one or two books in the last month?Did you read in the last month?How many books did you read in the last month?Answer:
How many books did you read in the last month?
Step-by-step explanation:
A tally chart is a numbering system used to quickly record data. It usually involves filling a chart with vertical dashes that represents the number of times an action or information is observed.
Therefore, by asking, "How many books did you read in the last month?", Alexandra would be able to get the answer by means of a tally chart which would show the number of times that action occurred.
–5(–2) =
whats the answer
Answers
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
What is the opposite of -2.3
Answers
Answer:
2.3
Step-by-step explanation:
because its asking you what the opposite of -2.3 is so it would be a positive instead of a negative
Answer:
The answer would be 2.3
The opposite of a negative is a positive, and vice versa
Step-by-step explanation:
Hope this helps
Kelley has to work 45 hours at her job to earn enough money to buy a phone. She works 3 hours each day. If she has worked x days, write a linear equation to represent how many more hours Kelley must work to afford the phone.
Answers
The linear equation that represents how many more hours Kelley must work to afford the phone is 3x - 45 = 0
Functions and valuesFunctions are expressions represented in terms of a variable.
From the question, we have the following
Total number of hours = 45
If she works 3 hours per day and has worked x days, the total number of hours she needs to work is 3x
Equating to the total hours wwill give:
3x = 45
3x - 45 = 0
Hence the linear equation that represents how many more hours Kelley must work to afford the phone is 3x - 45 = 0
Learn more on linear equation here: https://brainly.com/question/14323743
ordered pair 0,0 is also known as the
Answers
Hey there!
Please remember that the ordered pair (0,0) is also called the origin.
Its x-coordinate is 0, and its y-coordinate is 0.
If a line intersects a point (0, 0) then it basically intersects the origin.
Hope everything is clear.
Let me know if you have any questions!
#LearningDoesn'tEnd
:-)
I need help with this problem
Answers
a cyclist bike a certain distance in 45 minutes. How long will it take a pedestrian to travel the same distance if he travels 2 2/3 times slower than the cyclist in minutes to hours.
Need by 3/14/22
Answers
Multiply time by the difference:
45 x 2 2/3 = 120 minutes
1 hour = 60 minutes
120/60 = 2 hours
Answer: 120 minutes which equals 2 hours
You plant a tomato plant in your garden. It starts at 8 inches tall. After 1 week, it grows to 11 inches tall. After two weeks, it is now 14 inches tall. After 3 weeks, the plant is 17 inches tall. Write a linear equation that represents this model. How tall is the tomato plant after 56 weeks?
Answers
Answer:
The tomato plant is 176 inch tall after 56 weeks.
Step-by-step explanation:
x ⇒ weeks and y ⇒ inch.
Week 0 the tomato plant starts at 8 inch tall, week 1 it grows to 11 inch tall, week 2 it grows to 14 inch tall, week 3 it grows to 17 inch tall, week 56 is ? inch tall. So, every week (x) the tomato plant grows 3 inch (y).
y = 3x + 8
y = (3 × 56) + 8
y = 168 + 8
y = 176 inch
[tex]h(n) = 8 + 3n[/tex]
Using the function, after 56 weeks, the height of the plant is of 176 feet.The linear model for the height after n weeks has the following format:
[tex]h(n) = h(0) + an[/tex]
In which:
h(0) is the initial height. a is the rate of change.Starts at 8 inches tall, thus h(0) = 8. Each week the height increases 3 inches, thus, a = 3, and the model is:
[tex]h(n) = 8 + 3n[/tex]
After 56 weeks, the height is of h(56), thus:
[tex]h(56) = 8 + 3(56) = 176[/tex]
176 inches tall after 56 weeks.
A similar problem is given at https://brainly.com/question/16302622
Olivia purchases gifts for her mom and grandma for mothers day. the first is a pair of sunglasses for 16.35 and the second is a bracelet. her total is 39.75. how much did she spend on the bracelet?
Answers
23.40 because 39.75 minus 16.35 equals 23.40 Hope this helps
m = -(4+ m) +2
what is m?
Answers
Answer:
m = -1
Step-by-step explanation:
[tex]m = - (4 + m)[/tex]
Use - to open the bracket
[tex]m = - 4 - m + 2[/tex]
Collect like terms and simplify
[tex]m + m = - 4 + 2 \\ 2m = - 2[/tex]
Divide both sides of the equation by 2
[tex] \frac{2m}{2} = \frac{ - 2}{2} [/tex]
Simplify
[tex]m = - 1[/tex]
Write the polynomial
Answers
Answer:
17x^5 + 5x³ + 4x - 7
Degree of polynomial: 5
Step-by-step explanation:
First put the polynomial is descending order from highest exponent to lowest.
Original: 4x + 5x³ + 17x^5 - 7
New: 17x^5 + 5x³ + 4x - 7
The highest degree is 5 because that's the highest exponent. If it were m²x³ + 7x. The highest degree would be 5 because the exponents equal 5.
I hope this helps!
Write an expression for the area of a square with sides of length a + b.
Use the distributive property to fully expand the expression.
Show your work.
Area
Answers
Answer:
area = (a + b)² = a² + 2ab + b²
Step-by-step explanation:
let side length = a + b
area of a square = side length × side length
⇒ area = (a + b)²
= (a + b)(a + b)
= a² + ab + ab + b²
= a² + 2ab + b²
The fully expanded expression for the area of a square with sides of length a + b is a² + 2ab + b².
The area of a square is given by the formula: Area = side × side.
In this case, the side of the square is a + b.
Therefore, the expression for the area of the square is:
Area = (a + b) × (a + b).
To fully expand this expression using the distributive property, we'll apply the FOIL method (First, Outer, Inner, Last):
Area = a(a + b) + b(a + b)
= a² + ab + ab + b²
= a² + 2ab + b².
So, the fully expanded expression for the area of a square with sides of length a + b is a² + 2ab + b².
To learn more on Expression click:
brainly.com/question/14083225
#SPJ3
a formula for 6 gallons of light green paint uses 3/8 gallon of white paint. Liam has 9/16 gallon does Liam have enough white paint to make 8 gallons of light green paint?
Answers
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.
Solution:
Here, 6 gallons of light green uses [tex]\dfrac{3}{8}[/tex] gallon of white paint.
So, the white paint for 1 gallon of light green [tex]=\dfrac{\dfrac{3}{8}}{6}[/tex]
[tex]=dfrac{3}{8\times 6}[/tex]
[tex]=\dfrac{3}{48}[/tex]
[tex]=\dfrac{1}{16}[/tex]
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]
[tex]=\dfrac{8}{16}[/tex]
She has [tex]\dfrac{9}{16}[/tex] gallon of white paint, which is greater than [tex]=\dfrac{8}{16}[/tex] gallons.
Therefore, Liam have enough white paint to make 8 gallons of light green paint.
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
2x = 18
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
Learn more here: https://brainly.com/question/3796978
help please.! and thank you
Answers
Answer:
common difference is 4
Step-by-step explanation:
12-4=8
8-4=4
Helps me solve this problem please middle one
Answers
Answer:
D
Step-by-step explanation:
Hope u got it.......
3( x - 4) = -2( 4 - x)
Answers
3x-2x=-8+12
X=4
Please make me brainlist.
Step-by-step explanation:
first multyply x-4 by 3
second: same as first no:
put the algebric expression in same side
then do 3x-2x = -8 plus 12
then your answer is ready
Roger had 2kg of rice, he sold 15%.How many grams of rice did he sell
FULL EXPLANATION AND WORKING
Answers
Answer:
300 grams
Step-by-step explanation:
2kg = 2000 grams
15% = 15/100
15/100 × 2000 = 300 grams
PLEASE MARK ME AS BRAINLIEST
Which exponential function is equivalent to the geometric sequence an
= 7.5 (3)(n-1);
Answers
Answer:
f(x)=3^x
Step-by-step explanation:
assuming the n-1 is the exponent,
f(x)=3^x
the -1 is removed due to a graph starting at 0
On your graph paper, plot the two points. Then determine the distance between the two points. Round your answers to the nearest hundredth. (2,3) and (-1,4 )
Answers
Answer:
3.16
Step-by-step explanation:
estimate the area of the circle. use 3.14 3.14 or 22 7 227 for π π.
Answers
Answer:
615.44
Step-by-step explanation:
If the diameter is 28, that means the radius is 14.
The formula for the area of a circle is pi x radius to the power of 2. So, 14 to the power of 2 (or 14 x 14) equals 196. 196 x 3.14 (pi) = 615.44
The estimated area of the circle is about 615.44 cm² or 616 cm², depending on which value you use for π.
How to find the area of a circle?Suppose the circle has radius of 'r' units, then, its area is given as:
[tex]A = \pi \times r^2[/tex] sq. units
Since radius of a circle is half of its diameter, so if diameter is of 'd' length, then r= d/2, thus, area can be rewritten as:
[tex]\pi \times (\dfrac{d}{2})^2[/tex] sq. units
We are given that;
Diameter=28cm
Now,
In this case, the diameter of the circle is 28 cm, so the radius is half of that:
r = 28 / 2 = 14 cm
Plugging this value into the formula, we get:
A = π × 14²
Now we can use either 3.14 or 22/7 for π. Let’s use both and compare:
A ≈ 3.14 × 14² ≈ 3.14 × 196 ≈ 615.44 cm²
Or
A ≈ (22/7) × 14² ≈ (22/7) × 196 ≈ (4312/7) cm² ≈ 616 cm²
Therefore, by the area of circle the answer will be 615.44 cm² or 616 cm², depending on which value you use for π.
Learn more about area of a circle here:
https://brainly.com/question/13004063
#SPJ3
Someone help pls. It would mean a lot to me
Answers
Answer:
Step-by-step explanation:
A traffic helicopter descends 107 meters to be 407 meters above the ground, as illustrated in the diagram to the right. Write an equation to describe the situation using h as the original height of the helicopter. What was the original height of the helicopter?
Answers
Step-by-step explanation:
Its 514 because 107+407 is 514
Answer:
h = 514 meters
Step-by-step explanation:
If the helicopter descends 107 meters to then be 407 meters above the ground, then it was originally 107 meters above 407 meters.
h-107=407
add 107 to both sides
h = 107+407
h = 514
Heather is 5 feet 7 inches tall. There are
approximately 2.5 cm in one inch. What is
Heather's height in centimeters?
Answers
5’7” is equal to 67 inches
when you multiply 67 inches by 2.5 cm (which is around 1 inch) you get 167.5cm