Final answer:
To find the length of each leg of an isosceles triangle with an area of 125 ft² and a base of 16 ft, we first calculate the height and then apply the Pythagorean Theorem. The length of each leg is approximately 17.6 ft.
Explanation:
The question asks about finding the length of each leg of an isosceles triangle given the area and the base. To solve this, we use the formula for the area of a triangle, A = 1/2 × base × height, and apply the Pythagorean Theorem since the height creates a right triangle with the leg of the isosceles triangle and half of the base.
First, we find the height using the given area, A = 125 ft² and base, b = 16 ft. The formula rearranges to height = 2A / base, giving us height = 2(125) / 16 = 15.625 ft.
Next, using the Pythagorean Theorem, a² + b² = c², we let a represent half of the base (8 ft), b be the height (15.625 ft), and c be the leg of the triangle. Solving for c, we find c ≈ 17.6 ft, rounded to the nearest tenth.
Find r(t) if r'(t) = 3t2i + 4t3j + t k and r(1) = i + j.
Final answer:
To find the position vector r(t) given its derivative and an initial condition, we integrate each component of the derivative and apply the initial condition to solve for constants of integration. The final position function is r(t) = t³i + t⁴j + (0.5t² - 0.5)k.
Explanation:
To find r(t) given that r'(t) = 3t²i + 4t³j + tk and r(1) = i + j, we integrate each component of r'(t) with respect to t. The integral of a derivative returns the original function plus a constant of integration, which we can solve using the initial condition provided.
For the i component: ∑ 3t² dt = t³+ C1
For the j component: ∑ 4t³ dt = t⁴ + C2
For the k component: ∑ t dt = 0.5t² + C3
Applying the initial condition r(1) = i + j, we substitute t = 1 into r(t) to solve for the constants of integration:
(1) + C1 = 1, so C1 = 0
(1) + C2 = 1, so C2 = 0
0.5(1) + C3 = 0, so C3 = -0.5
Therefore, the position function r(t) is given by t³i + t⁴j + (0.5t² - 0.5)k.
r(t) = t^3i + t^4j + ((1/2)t^2 - 1/2)k.
To find r(t) given r'(t) = 3t^2i + 4t^3j + tk and r(1) = i + j, we need to integrate r'(t) with respect to t.
Step 1: Integrate each component of r'(t) separately:
∫3t^2 dt = t^3 + C1 (integration with respect to t)
∫4t^3 dt = t^4 + C2 (integration with respect to t)
∫tk dt = (1/2)t^2k + C3 (integration with respect to t)
Step 2: Combine the results to get r(t):
r(t) = (t^3 + C1)i + (t^4 + C2)j + ((1/2)t^2k + C3)
Step 3: Use the given initial condition r(1) = i + j to find the values of the constants C1, C2, and C3:
r(1) = (1^3 + C1)i + (1^4 + C2)j + ((1/2)(1)^2k + C3)
i + j = i + j + (1/2)k + C3
Comparing the coefficients of k, we get:
(1/2) + C3 = 0
C3 = -1/2
Therefore, r(t) = (t^3 + C1)i + (t^4 + C2)j + ((1/2)t^2 - 1/2)k
Please help:
Solve using logarithms. Round the solution to the nearest hundredth. Show your work.
(you will need to use a calculator at some point)
A gear rotates one degree each second. If it rotates for 2 minutes, how many degrees will it measure
Keisha's boat has a top speed of 9 miles per hour in still water. While traveling on a river at top speed, she went 10 miles upstream in the same amount of time she went 20 miles downstream. Find the rate of the river current.
Jenna has 8 pairs of jeans. Three of them are blue. What percentage of her jeans is blue?
= 37.5%
But my question is this:
Jenna bought two more pairs of jeans.
If both of them are blue, what percentage of her jeans is blue now?
If both of them are not blue, what percentage of her jeans is blue now?
Which equation represents a nonlinear function?
x(y – 5) = 2
y – 2(x + 9) = 0
3y + 6(2 – x) = 5
2(y + x) = 0
Answer:
The correct answer is: A: x(y – 5) = 2
Step-by-step explanation:
edge2021
What is a sequence? a sequence is an unordered list of numbers. a sequence is an ordered list of numbers. a sequence is the sum of an?
You have an equally likely chance of choosing any number from 1 to 10. What is the probability that you choose a number greater than 6?
Answer:
2/5
Step-by-step explanation:
1 to 10, it is 10 numbers
and there are 4 numbers which are 7,8,9,10 are bigger than 6
so P =4/10=2/5
Leo painted for 2/3 of an hour each day on Monday, Tuesday, Thursday and Friday. How long did Leo paint this week
Leo painted for 2/3 of an hour each day on Monday, Tuesday, Thurday, and friday. Thus, the total time of the week for which he painted is 1.58% of the total time.
What is a fraction?A fraction represents a part of a whole number or, more commonly, any number of equal parts. A fraction can be defined as the parts of a certain size, for example, one-half, eight-fifths, three-quarters, etc.
The time for which Leo painted in this week can be calculated as:
Leo painted 2/3 of an hour each day for monday, tuesday, thursday, and friday
Therefore, the time of painting in an hour = 40 minutes
2/3 × 60 = 120/ 3 = 40 minutes = 40 × 4 = 160 minutes
The number of minutes in a day,
1 day = 60 × 24 = 1,440 minutes
The number of minutes in seven days,
7 days = 1440 × 7 = 10,080 minutes
Therefore, the total time for which Leo painted = time for which he painted in minute divided by total time in minutes
Total time for which Leo painted = 160/ 1440 = 0.0158 or 1.58%
Therefore, total time for which he painted in this week is 0.0158 of total time
Learn more about Fraction here:
https://brainly.com/question/10354322
#SPJ5
2.7 Q9
The functions f and g are defined by the following tables. Use the tables to evaluate the given composite function.
Answer: [tex](gof)(-1)=-2[/tex]
Step-by-step explanation:
By definition, a composite function is a function formed by substituing one function into another function.
Given the composite function:
[tex](gof)(-1)[/tex]
You can rewrite it in the following form:
[tex]g(f(-1))[/tex]
In order to evaluate it, follow these steps:
- Based on the table of the function f(x), the ouput value for [tex]x=-1[/tex] is [tex]y=3[/tex]. Then:
[tex]f(-1)=3[/tex]
- This will be the input of the function g(x).
- Notice in the table of the function g(x) that the output value when [tex]x=3[/tex] is [tex]y=-2[/tex]:
[tex]g(3)=-2[/tex]
Therefore:
[tex]g(f(-1))=-2[/tex] or [tex](gof)(-1)=-2[/tex]
Answer:
The value of [tex]\rm{gof(-1)}[/tex] is -2.
Step-by-step explanation:
Given,
The table contains the functional value of f(x) and g(x).
To find: [tex]\rm{gof(-1)}[/tex]
[tex]\rm{fog(-1)}[/tex] is a composite function of f(x) and g(x).
[tex]\rm{gof(-1)}=\rm{g[f(-1)}][/tex]
Clearly from the table, the value of f(x) when [tex]x=-1[/tex] is 3.
Therefore,
[tex]f(-1)=3[/tex]
Now,
[tex]\rm{gof(-1)}=\rm{g[f(-1)}]\\\rm{fof(-1)}=\rm{g(3)}[/tex]
Clearly from the table, the value of g(x) when [tex]x=3[/tex] is -2.
Thus,
[tex]\rm{gof(-1)}=-2[/tex]
To know more, please refer to the link:
https://brainly.com/question/18839694?referrer=searchResults
Consider the diagram and the paragraph proof below. Given: Right △ABC as shown where CD is an altitude of the triangle Prove: a2 + b2 = c2 Because △ABC and △CBD both have a right angle, and the same angle B is in both triangles, the triangles must be similar by AA. Likewise, △ABC and △ACD both have a right angle, and the same angle A is in both triangles, so they also must be similar by AA. The proportions and are true because they are ratios of corresponding parts of similar triangles. The two proportions can be rewritten as a2 = cf and b2 = ce. Adding b2 to both sides of first equation, a2 = cf, results in the equation a2 + b2 = cf + b2. Because b2 and ce are equal, ce can be substituted into the right side of the equation for b2, resulting in the equation a2 + b2 = cf + ce. Applying the converse of the distributive property results in the equation a2 + b2 = c(f + e).
Answer:
Because f+e=c
Therefore ,[tex]a^2+b^2=c^2[/tex]
Step-by-step explanation:
Given A right triangle ABC as shown in figure where CD is an altitude of the triangle.
To prove that [tex]a^2+b^2=c^2[/tex]
Proofe: Given [tex]\triangle ABC\; and\; \triangle CBD[/tex] both are right triangle and both triangles have common angle B si same.
Therefore , two angles of two triangles are equal .
Hence, [tex]\triangle ABC \sim\triangle CBD[/tex] by using AA similarity.
Similarity property: when two triangles are similar then their corresponding angles are equal and their corresponding side are in equal proportion.[tex]\frac{a}{f} =\frac{c}{a}[/tex]Similarly , [tex]\triangle ABC \sim \triangle ACD[/tex] by AA similarity property . Because both triangles are right triangles therefore, one angle of both triangles is equal to 90 degree and both triangles have one common angle A is same .
[tex]\therefore[/tex] [tex]\frac{b}{c} =\frac{e}{b}[/tex]
The corresponding parts of two similar triangles are in equal proportion therefore , two proportion can be rewrite as
[tex]a^2=cf[/tex] (I equation)
and [tex]b^2=ce[/tex] (II equation)
Adding [tex]b^2[/tex] to both sides of firs equation
[tex]a^2+b^2=cf+b^2[/tex]
Because [tex]b^2=ce[/tex] and ce can be substituted into the right side of equation wevcan write as
[tex]a^2+b^2=ce+cf[/tex]
Applying the converse of distributive property we can write
[tex]a^2+b^2=c(f+e)[/tex]
Distributive property: a.(b+c)= a.c+a.b
[tex]a^2+b^2=c^2[/tex]
Because f+e=[tex]c^2[/tex]
Hence proved.
Jayden has been reading a 275275275-page book for English class. On average, he has read about 101010 pages in 151515 minutes (\text{min})(min)left parenthesis, m, i, n, right parenthesis. With a deadline looming, Jayden realizes that he won't finish in time unless he skims through the rest of the book. If he skims the book, he can cover 151515 pages in 10\,\text{min}10min10, space, m, i, n. If it takes Jayden 555 hours and 50\,\text{min}50min50, space, m, i, n to finish the book (reading and skimming combined), how many pages did Jayden have to skim through?
What is the range of the following set?
28, 45, 12, 34, 36, 45, 19, 20
8
33
45
57
The circle is centered at the point (-3,4) and has a radius of length 3. What is its equation?
Container A can hold 9 cups of liquid. container B can hold 5 pints of liquid. Container c can hold 2 quarts of liquid. Whice container has the greatest capacity? Explain
the second of two numbers is 4 more than the first the sum of the numbers is 56 find the numbers
Answer:
26, 30
Step-by-step explanation:
If you subtract from the sum the excess of the second over the first, you have a sum that is twice the first.
first number = (56 -4)/2 = 26
The second number is 4 more, so is ...
second number = 26 +4 = 30
_____
Check
Their total is 26 +30 = 56, as required.
Use the chain rule to find ∂z/∂s and ∂z/∂t. z = x5y7, x = s cos(t), y = s sin(t)
Using the Chain Rule of differentiation, we can solve for ∂z/∂s and ∂z/∂t given the functions for z, x, and y. The Chain Rule is fundamental for calculus by finding the derivative of composite functions.
Explanation:To solve for ∂z/∂s and ∂z/∂t with z = x5y7, x = s cos(t), and y = s sin(t), we'll employ the use of the Chain Rule of differentiation. First, we consider ∂z/∂s. As z is x raised to the power of 5 then multiplied by y to the power of 7, we can express ∂z/∂x and ∂z/∂y. Proceeding using the chain rule, which in simpler terms could be stated as 'the derivative of the outside times the derivative of the inside', we get the values of ∂z/∂s and ∂z/∂t. The chain rule in calculus is an important tool when differentiating composite functions.
Learn more about Chain Rule here:https://brainly.com/question/28972262
#SPJ12
Reciprocal of 1/10 = ?
Reciprocal of 16 = ?
Please help me with question 22
Find the longer diagonal of a parallelogram having sides of 10 and 15 and an angle measure of 120° between them. 13.2 18.0 21.8 23.1
Answer:
21.8
Step-by-step explanation:
A roadway repair crew has fixed 42 mi of the roadway. That number of miles is 30% of the total number of miles that need to be repaired
Answer:
The answer is 140 miles 100x42 is 4200 divide that by 30= 140Step-by-step explanation:
A band that will be the opening act for a concert charges a $500.00 flat fee as well as 10% of all profits from ticket sales. If the band earns $1,200, how much was made in total ticket sales for the show?
Answer:
The answer is 7,000
Your welcome ;)
consider the quadratic equation x^2=4x-5.How many solutions does the equation have?
paul is in charge of the egg toss at the world egg day fair. there are 48 people participating in teams of 8 people. each team needs 1 egg. paul is buying eggs in cartons that each have 6 eggs. how many cartons does paul need?
Final answer:
Paul needs to buy 1 carton of eggs, as there are 6 teams needing 1 egg each, and one carton contains 6 eggs.
Explanation:
Paul needs to calculate how many cartons of eggs to buy for the egg toss at the world egg day fair.
With 48 people participating in teams of 8, we have a total of 48/8 = 6 teams, and each team needs 1 egg.
Since each carton contains 6 eggs, Paul needs to buy 6/6 = 1 carton to provide one egg for each team.
The boys' basketball team at Washington High School has been doing really well so far this year. In the past six games, they've scored the following points.
84, 67, 74, 87, 67, 73
What is the mean of their scores, rounded to the nearest tenth?
Answer:
(84 +67 +74 +87 +67 +73)/6 = 452/6 = 75.3
Step-by-step explanation:
Exactly what you see :]
This figure consists of a rectangle and semicircle.
What is the perimeter of this figure?
Use 3.14 for pi.
24.00 ft
30.28 ft
34.28 ft
36.56 ft
HELP ASAP WILL GET BRAINLIEST AND 10 POINTS!! IF CORRECT
Answer:
30.28
Step-by-step explanation:
just did the k12 test so i thought i would help out:)
12km is equal to how many meters? (show all work)
This week laptops are 25 percent off if a laptop cost $340 what is it's sale price
BRAINLIEST!!!!!!!
Quadrilateral ABCD is inscribed in this circle.
What is the measure of angle C?
Enter your answer in the box.
Answer:
92 is correct
Step-by-step explanation:
convert the following logarithmic equation to the equivalent exponential equation. Use the caret (^) to enter exponents. y=ln x
Answer:
To solve this problem, you need to remember that an exponential function has the following form:
f(x)=a^x
"a" is the base and "x" is the exponent.
2. It is important to know that the logarithmic functions and the exponential functionsare inverse. Then, you have:
y=ln x
e^y=e^(lnx)
e^y=x
3. Therefore, the answer is:
Step-by-step explanation: