Final answer:
The length of the median of the trapezoid with perpendicular diagonals of lengths 8 and 10 is 9 units. This is calculated using properties of the median and the Pythagorean theorem.
Explanation:
The question you've asked regarding the median of a trapezoid with perpendicular diagonals of lengths 8 and 10 can be resolved by recognizing a property of the median in a trapezoid. The median (also known as the mid-segment) of a trapezoid is parallel to the bases and its length is equal to the average of the lengths of the bases. Since the diagonals are perpendicular, they would form right triangles with the two bases and the median line, dividing the trapezoid into four right triangles.
Let's denote the lengths of the bases as a and b, and the median as m. We know the diagonals intersect at their midpoints, thus splitting each into segments of lengths 4 and 5. Now we can form two right triangles sharing the median as a common side. Applying the Pythagorean theorem, we get two equations: m² + 4² = a² for the first right triangle and m² + 5² = b² for the second right triangle.
Since the median is the average of the bases, we have m = (a + b) / 2. Using the equations above, after some algebraic manipulations, we find that a² - b² = 16. With more manipulation, eventually, we find that (a - b)(a + b) = 16, and, since m is the average, 2m = a + b. Hence, we derive that 2m - b = (a - b), which simplifies to give us m = 9. This gives us the length of the median of the trapezoid as 9 units.
The French club is sponsoring a bake sale. If their goal is to raise at least $135, how many pastries must they sell at $2.50 each in order to meet that goal? Write and solve an inequality.
Last year, there were n pies baked for the bake sale. This year, there were 156 pies baked. Using n, write an expression for the total number of pies baked in the two years
How do I simplify this problem?
A sum of money amounting to $4.25 consists of dimes and quarters if there are 26 coins in all how many are quarters
d= dimes
q = quarters
d+q=26 coins
rewrite as d=26-q
0.25q +0.10d=4.25
0.25q+0.10(26-q)=4.25
0.25q+2.6-0.10q=4.25
0.15q=1.65
q=11
d=26-11=15
11*0.25 = 2.75, 15*0.10=1.50, 2.75+1.50=4.25
there are 11 quarters
Planes Q and R are parallel. Explain how you know lines a and b are skew.
Sample Response: Skew lines are noncoplanar and do not intersect. Line a lies in plane Q and line b lies in plane R, so the lines are not coplanar. No other plane can be drawn through the lines, so they are not parallel. So, a and b are skew
Sample Response:
Skew lines are noncoplanar and do not intersect. Line a lies in plane Q and line b lies in plane R, so the lines are not coplanar. No other plane can be drawn through the lines, so they are not parallel. So, a and b are skew.
Please show work on how you got the answer
Is #7 correct? Please explain.
Solve x3 = 64 over 27.
±8 over 3
8 over 3
±4 over 3
The solution of the given equation is ±8/3.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
The solution of an equation is the set of all values that, when substituted for unknowns, make an equation true.
The given equation is x³= 64/27
x=±∛(64/27)
x=±∛(8³/3³)
x=±8/3
Therefore, the solution of the given equation is ±8/3.
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If n is a positive integer and the product all the integeres from 1 to n inclusive is a multiple of 990 what is the least possible value of n
PLEASE HELP ON THESE ILL GIVE 20 POINTS AND A BRAINLIEST IF YOUR CORRECT!!
Variation is a term that is used to describe __________.
A.
how repetitive a data set is
B.
how large or small a data set is
C.
how spread out or scattered a data set is
D.
how different a data set is from other data sets
The function P(t) = 5,000 (0.93)t is the value of a motor scooter t years after it is purchased. Which of the following values show the annual depreciation of the value of the motor scooter?
93
-7
50
7
Answer:
Option 4 - 7%
Step-by-step explanation:
Given : The function [tex]P(t) = 5000 (0.93)^t[/tex] is the value of a motor scooter t years after it is purchased.
To find : Which of the following values show the annual depreciation of the value of the motor scooter?
Solution :
We are given a function p(t) which is represented as:
[tex]P(t) = 5000 (0.93)^t[/tex]
Where, p(t) is the value of a motor scooter t years after it is purchased.
Depreciation is the amount that has been deducted from the cost of an item.
The rate of change is 1-0.93=0.07 i.e. 7%.
So, The function P(t) has 7% of the amount has been deducted each year and the value of the motor scooter is 93% of the amount of the previous year.
Therefore, The annual rate of depreciation is 7%.
So, option 4 is correct.
What is the probability that when a fair coin is flipped 25 times, there will be exactly five heads
What are the phase shift and period for the function y = 3cos[4(θ + 60°)] − 2?
Phase shift = right 60°, period = 90°
Phase shift = right 60°, period = −90°
Phase shift = left 60°, period = 90°
Phase shift = left 60°, period = −90°
PLEASE HELP!!!!!!!
f(x)=x^2−3x+9
g(x)=3x^3+2x^2−4x−9
Find (f−g)(x)
Select one:
a. −3x^3−x^2+x+18
b. 3x^3+3x^2−7x
c. 3x^3+x^2−x−18
d. 3x^3−x^2−x
cot^2x-csc^2x=-1 for all values of x true or falsse
The original equation cot^2x - csc^2x = -1 is true for all values of x.
How to determine if cot^2x-csc^2x=-1 for all values of xThe equation cot^2x - csc^2x = -1 is true for certain values of x, but not for all values of x.
To see why, let's break down the equation using trigonometric identities:
cot^2x - csc^2x = -1
Using the reciprocal identities, we can rewrite cot^2x and csc^2x in terms of sine and cosine:
(cos^2x / sin^2x) - (1 / sin^2x) = -1
Now, let's simplify:
(cos^2x - 1) / sin^2x = -1
Using the Pythagorean identity cos^2x + sin^2x = 1, we can substitute cos^2x with (1 - sin^2x):
(1 - sin^2x - 1) / sin^2x = -1
-sin^2x / sin^2x = -1
Now, we can cancel out the sin^2x terms:
-1 = -1
This equation holds true for all values of x. Therefore, the original equation cot^2x - csc^2x = -1 is true for all values of x.
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Find two positive numbers such that their product is 192 and their sum is a minimum pre-calc
The two numbers with a product equal to 192 such that their sum is minimized are:
A = √192 and B = √192
How to find the two numbers?
Let's define A and B as our two numbers, we know that their product must be equal to 192, then we have:
A*B = 192.
Now, the sum of these two numbers is:
A + B.
And we want to minimize this, to do it, we need to use the first equation to rewrite one variable in terms of the other. For example, if we isolate A, we get:
A = 192/B
Replacing this in the sum, we get:
192/B + B
To minimize this, we need to find the values of B that make 0 the differentiation of the above expression.
The differentiation is:
-192/B^2 + 1
Then we need to solve:
-192/B^2 + 1 = 0
192/B^2 = 1
192 = B^2
√192 = B
To get the value of A, we use:
A = 192/B = 192/√192 = √192
Then we can conclude that the two positive numbers such that their product is 192, and their sum is minimized, is:
A = √192 and B = √192.
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How many different selections of 4 books can be made from a bookcase displaying 12 books?
(12 x 11 x10 x9) / (4 x3 x 2 x 1) = 495
there are 495 different ways to select 4 books
The law of cosines is a^2+b^2-2abcos(C). Find the value of 2abcos(C).
A. 37
B. -40
C. 40
D. 20
The length of a rectangular lawn is measured to twice its width. the perimeter of the lawn is given as 30 m. find the length and width of the lawn.
18 divided by 5593 equals 31-5400 = 193 what number should be placed in the box to help complete the division calculation?
The number that should be placed in the box to complete the division calculation is -7.
To find the number that should be placed in the box to complete the division calculation, let's analyze the given information:
- Dividend: 5593
- Divisor: 18
- Quotient: 31 (hundreds place is 3 and tens place is 1)
- When 5400 is subtracted from 5593, the result is 193.
- The box represents an unknown number in the ones place of the quotient.
To find the value in the box, we need to consider the units place. Since 5400 is subtracted from 5593 to give 193, the difference between the units places should be the value in the box.
193 - 5400 = -5207
So, the number that should be placed in the box to complete the division calculation is -7.
Correct question is:
What number should be placed in the box to help complete the division calculation?
Long division setup showing an incomplete calculation. 18 is in the divisor, 5593 is in the dividend, and 3 hundreds and 1 tens is written in the quotient.
5400 is subtracted from 5593 to give 193.
An unknown value represented by a box is being subtracted from 193.
Which sample fairly represents the population? CHECK ALL THAT APPLY.
--measuring the heights of every fiftieth person on the school roster to determine the average heights of the boys in the school
--calling every third person on the soccer team’s roster to determine how many of the team members have completed their fundraising assignment
--observing every person walking down Main Street at 5 p.m. one evening to determine the percentage of people who wear glasses
--sending a confidential e-mail survey to every one-hundredth parent in the school district to determine the overall satisfaction of the residents of the town
--taking a poll in the lunch room (where all students currently have to eat lunch) to determine the number of students who want to be able to leave campus during lunch
Answer:
The population is defined by: The set of all possible individuals who are likely to be selected for a study and who also share some observable characteristic. In this way, the correct answer is:
Taking a poll in the lunch room (where all students currently have to eat lunch) to determine the number of students who want to be able to leave campus during lunch
Step-by-step explanation:
Bicycle city makes custom bicycles. They charge $160 plus $80 for each day that it takes to build the bicycle. If you have $480 to spend on your new bicycle, how many days can it take Bicycle City to build the bike?
Find the value of n in the equation 6.2n – 3.7n = 85 + 45. A. 16 B. 52 C. 13.13 D. 325
The tip of an 11-inch wiper blade wipes a path that is 31 inches long. What is the angle of rotation of the blade in radians to the nearest tenth?
2.8 radians is the answer.
The sum of four consecutive even integer numbers is 84. find the four numbers.
84/4=21
now take the 2 even numbers below 21 and the 2 even numbers above 21
18 +20 + 22 +24 = 84
the numbers are 18, 20, 22 & 24
Prove that if one solution for a quadratic equation of the form x 2 + bx + c = 0 is rational (where b and c are rational), then the other solution is also rational. (use the fact that if the solutions of the equation are r and s, then x 2 + bx + c = (x − r)(x − s).)
Final answer:
If one root of a quadratic equation with rational coefficients is rational, the other root must be rational too because the sum and product of the roots are related to the coefficients, which are also rational.
Explanation:
To prove that if one solution for a quadratic equation of the form x^2 + bx + c = 0 is rational, then the other solution is also rational, we can use the quadratic formula and properties of rational numbers. If the quadratic equation has rational coefficients and one rational solution, then the sum and product of the roots must also be rational. This is because a quadratic equation with roots r and s can be factored as (x - r)(x - s) = 0, which expands to x^2 - (r + s)x + rs = 0. Matching coefficients, we see that - (r + s) = b and rs = c. Since b and c are rational, r + s and rs must be rational as well.
Given that we have one rational root, let's say r, the sum of the roots r + s is rational, so s must also be rational because the difference of two rational numbers is rational. Hence, if one root of a quadratic equation with rational coefficients is rational, the other root must be rational as well.
Adam has $450. he spends $210 on food. later he divides all the money into four parts out of which three parts were distributed and one part he keeps for himself. then he found $50 on the road. write the final expression and find the money he has left?
Solve log2(6-2x)-log2x=3
The following two-way table shows the number of students of a school who have a video game and/or have a laptop:
Have Video Game Do Not Have Video Game Total
Have Laptop
15
50
65
Do Not Have Laptop
50
15
65
Total
65
65
130
Based on the table, how many students have both a laptop and a video game?
15
50
65
130
Based on the table given, the number of students who have both a laptop and a video game is 15 students.
Which students have both laptops and video games?In order to solve this question, look at the part of the table where the row on students who have laptops intersects with the column on those who have video games.
That part of the table has 15 in the cell. This means that 15 students have both video games and laptops.
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find the solution to the two equations shown below y=16x-9 y=-4x+21
Answer: (1.5, 15)
Step-by-step explanation:
The given system of equations :-
[tex]y=16x-9...................(1)\\\\y=-4x+21..................(2)[/tex]
Substituting the value of y from equation (1) in equation (2), we get
[tex]16x-9=-4x+21[/tex]
Now, Adding 4x and 9 on both the sides , we get
[tex]16x+4x=21+9\\\\\Rightarrow\ 20x=30\\\\\Rightarrow\ x=\dfrac{3}{2}=1.5[/tex]
Substitute the value of x in equation (1), we get
[tex]y=16(1.5)-9=15[/tex]
Hence, the solution to the given system of equations : (1.5, 15)
The solution to the given system of equation is (1.5, 15).
The given system of equations are
y=16x-9 ---------(i)
y=-4x+21 ---------(ii)
From equation (i) and (ii), we get
16x-9=-4x+21
16x+4x=21+9
20x=30
x=30/20
x=1.5
Substitute x=1.5 in equation (i), we get
y=16×1.5-9
y=24-9
y=15
Therefore, the solution to the given system of equation is (1.5, 15).
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