Let's denote the two consecutive positive even integers as x and x+2.
We know their squares sum up to 100, represented mathematically as:
x^2 + (x + 2)^2 = 100.
Expanding this equation, we get:
x^2 + x^2 + 4x + 4 = 100.
Combining like terms gives us:
2x^2 + 4x - 96 = 0.
This is a quadratic equation in the form of ax^2 + bx + c = 0, where a = 2, b = 4, and c = -96.
We can solve this quadratic equation for x using the quadratic formula:
x = [-b ± sqrt(b^2 - 4ac)] / 2a.
Substituting a, b, and c into this formula, we find that:
x = [-4 ± sqrt((4)^2 - 4*2*-96)] / 2*2,
x = [-4 ± sqrt(16 + 768)] / 4,
x = [-4 ± sqrt(784)] / 4,
x = [-4 ± 28] / 4.
So the solutions are x = (28 - 4) / 4 = 6 and x = (-28 - 4) / 4 = -8.
Since we are looking for positive even integers, we discard the negative solution.
Therefore, the two consecutive positive even integers are 6 and 6 + 2 = 8.
In how many ways can 3 identical emeralds 2 identical diamonds and 2 different opals be arranged in a row in a display case
They can be arranged in 210 ways.
We have given that, 3 identical emeralds 2 identical diamonds and 2 different opals
What is the arrangement of n object where n₁ is one kind, n₂ is a second kind, and n₃ is a third kind?The arrangement of n objects where n₁ is one kind, n₂ is a second kind, and n₃ is a third kind is given by
[tex]\frac{n!}{n_1!*n_2!*n_3!}[/tex]
The total number of gems =7
3 of the first kind, 2 of the second kind and 2 of the third kind
Therefore, we get
[tex]\frac{7!}{2!*2!*3!}=210[/tex]
Therefore ,they can be arranged in 210 ways.
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Bob bought 3 packs of model cars. He gave 4 cars to Ann.Bob has 11 model cars left how many model cars were in each pack.
Determine the values of x on which the function f(x)=2x^2-x-15/4x^2-12x is discontinuous and verify the type of discontinuity at each point.
A.There is a vertical asymptote at 0 and a hole at 3.
B.There are vertical asymptotes at -5/2 & 0 and a hole at 3.
C.There is a vertical asymptote at 3 and a hole at 0.
D.There are vertical asymptotes at 0 & 3.
There is a vertical asymptote at x = 0 (because the denominator becomes 0). We need to further analyze the behavior at x = 3 to see if it's a hole (cancellation) or the function is continuous there (Option A or B might be the answer).
Step 1: Identify Potential Discontinuities
A rational function (like f(x)) is discontinuous when its denominator is zero. So, we need to find the values of x that make the denominator 0:
Denominator = 4x^2 - 12x = 0
Factor the denominator: 4x(x - 3) = 0
This gives us two possible values for x: x = 0 and x = 3.
Step 2: Verify Discontinuity Type at Each Point
Check if the function is defined (has a value) at these points.
x = 0: If we plug x = 0 in the original function, we get 0/0, which is indeterminate. This suggests a potential vertical asymptote at x = 0.
x = 3: If we plug x = 3 in the original function, the expression doesn't become 0/0. The function might have a hole or be continuous at x = 3.
Step 3: Analyze for Holes (Optional):
To further investigate x = 3, we can try to simplify the function by factoring the numerator (possible because the denominator is already factored). If the simplification gets rid of the term that causes the 0/0 at x = 3, it creates a hole.
Drew ran 5 3/ 8 miles yesterday. Collin ran 8 5/12 miles yesterday. How much farther did Collin run? Choose the correct values to leave your answer as a mixed number. Numbers may be used once or not at all.
Answer:
The correct answer is 3 1/24
Step-by-step explanation:
because I did all the math
Answer:
3 1/24
Step-by-step explanation:
Suppose that the measure of the minor arc BD is 172°. Find m BCD. (I need help on all questions in picture)
Geometry A Problem! Help! Line AC
use the graph of y=sin theta to find the value of 2theta for theta = pi/4 radians
PLEASE PLEASE HELP ME WITH THESE!!!!
I REALLY NEED HELP AND I WILL AWARD BRAINLIEST FOR WHO IS CORRECT!!!
1. The table below shows the number of cameras sold in each month at the camera shop.
January: 35 cameras were sold
Feb. 21 cameras were sold
Mar. 35 were sold.
April: 55 were sold
May: 73 were sold
Jun. 82 sold
Jul. 85 were sold
Aug. 64 cameras were sold
Sep. 52 sold
Oct. 42 were sold
Nov. 89 were sold
Dec. 125 cameras sold
Use the set of data to answer the question.
The camera shop wants to use the measure of central tendency that expresses how successful they are in selling cameras. Which measure of central tendency should they choose?
A. Mean
B. Interquartile
C. Mode
D. Median
PLEASE HELP PLEASE AND THANK YOU!!!
can someone please help!!
Help please ! A B Or C
Answer:
B) 46
Step-by-step explanation:
What you have to do here is plug in the numbers 2, 3, 4, and 5 in for i in the equation and then add the sums together:
3(2) + 1 = 7
3(3) + 1 = 10
3(4) + 1 = 13
3(5) + 1 = 16
Add those together and they equal 46
Use graphing calculator to solve system the equations in the interval from 0+2pi.round to the nearest hundredth. 7cos 2t=3
The solutions of the trigonometric equation are approximately 0.56, 2.58, 3.71 and 5.72, all in concordance with the results found beside the analytical approach.
How to solve trigonometric equations
Trigonometric functions are periodic functions, cosine function has a standard period of 2π radians. There are two approaches to solve the expression given, an analytical one and a graphical one. Now we proceed to show each approach:
Analytical method
Now we proceed to apply algebraic handling and inverse trigonometric functions to solve the expression for t:
7 · cos 2t = 3
cos 2t = 3/7
2 · t = cos⁻¹ (3/7)
There are two solutions:
2 · t₁ ≈ 0.359π + 2π · i ∨ 2 · t₂ ≈ 1.641π + 2π · i
t₁ ≈ 0.180π + π · i ∨ t₂ ≈ 0.821π + π · i
t₁₁ ≈ 0.180π ∨ t₁₂ ≈ 1.180π ∨ t₂₁ ≈ 0.821π ∨ t₂₂ ≈ 1.821π
The solutions of the trigonometric equations in [0, 2π] are approximately 0.18π, 0.82π, 1.18π and 1.82π.
Graphical approach
First, we add the definition of the following two functions (f(x) = 7 · cos 2t and g(x) = 3) and plot in the calculator through a graphing tool. The solutions of this trigonometric function are the points of intersection of the two functions.
According to the image attached below, the solutions of the trigonometric equation are approximately 0.56, 2.58, 3.71 and 5.72, all in concordance with the results found beside the analytical approach.
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line AB is tangent to circle O at A. The diagram is not drawn to scale.
If AO=21 and BC=14, what is AB?
28 - 2(5a - 3) = 47
A. -1.3
B. 1.3
C. -2.5
D. 8.1
How much greater is the median number of pairs of shoes in the inventory at the Golf pro Shop than the median number of pairs of shoes in the inventory at the Tennis Pro Shop?
Identify the value that is NOT in the range of the piece-wise function.
A. -3
B. 1
C. 3
D. -1
A rancher has 4000 ft. of fence for constructing a rectangular corral that is divided into 3 equal sections. one side of the corral will be bounded by his barn and needs no fencing. what are the dimensions of the corral that will maximize the area? what is the area of the corral?
An angle measuring between 90° and 180° is called _____. A.obtuse B.acute C.straight D.small
hurry pls need help look at picture below
x 0 2 4 6 8 f(x) 15 15.5 17 19.5 23 Write the quadratic function that corresponds to the given table of values in standard form, f(x) = ax2 + bx + c, where a, b and c are constants.
f(x)=ax^2+bx+c
Answer with Step-by-step explanation:
We are given that:
f(x)=ax²+bx+c
also at x=0 f(x)=15
Putting x=0 in f(x)
a×0+b×0+c=15
⇒ c=15
now, f(2)=15.5
⇒ a×2²+b×2+c=15.5
⇒ 4a+2b+15=15.5
⇒ 4a+2b=0.5 -------(1)
f(4)=17
⇒ a×4²+b×4+c=17
⇒ 16a+4b+15=17
⇒ 16a+4b=2 --------(2)
equation (2)- 2×equation(1)
⇒ 16a+4b-8a-4b=2-2×0.5
⇒ 8a=2-1
⇒ 8a=1
⇒ a=1/8
⇒ a=0.125
Putting value of a and c in equation (1)
⇒ 4×0.125+2b=0.5
⇒ 0.5+2b=0.5
⇒ 2b=0
⇒ b=0
Hence
a=0.125
b=0
c=15
and f(x)=0.125x²+15
Write (3n)^3 without exponents
Answer:
Without exponents, we can write the term as: [tex]27(n)(n)(n)[/tex]
Step-by-step explanation:
The given term is :
[tex](3n)^{3}[/tex]
This can be written as:
[tex](3n)(3n)(3n)[/tex]
= [tex]27n^{3}[/tex]
= [tex]27(n)(n)(n)[/tex]
Hence, without exponents, we can write the term as: [tex]27(n)(n)(n)[/tex]
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Suppose you are headed toward a plateau 70 m high. If the angle of elevation to the top of the plateau is 60 degrees, how far are you from the base of the plateau?
To find the distance from the plateau's base, use the tangent function with the given height and angle of elevation.
Explanation:We can use trigonometry to determine the distance from the base of the plateau. The vertical height of the table (opposite side) is 70 m, and the elevation angle to the top of the plateau (theta) is 60 degrees. We can use the tangent function to find the distance (adjacent side) from the table's base: tan(theta) = opposite/contiguous. Solving for the space, we get: adjacent = contrasting/tan(theta). Plugging in the values, the distance from the plateau's base is approximately 40 m.
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Factor this expression completely and then place the factors in the proper location. Note: Place factors alphabetically!
mn - 4m - 5n + 20
To solve this problem you must apply the proccedure shown below:
1. You have the following expression given in the problem above:
[tex] mn - 4m - 5n + 20 [/tex]
2. The first thing you must do to factor the expression is to group the terms, as following:
[tex] (mn-4m)+(-5n+20) [/tex]
3. Now, choose the greatest common factor of each group:
[tex] m(n-4)-5(n-4) [/tex]
4. Now, you need to factor the expression by factoring out the [tex] n-4 [/tex]
[tex] (m-5)(n-4) [/tex]
The answer is: [tex] (m-5)(n-4) [/tex]
(6a −? )5a =? a2 − 35a
WHAT IS THE VALUE OF X
Solve 3/10 n-3 1/5 = -1 1/5
Answers-
A.n =-12 2/3
B.n = 6 2/3
C.n = 8
D.n = -6/25
Answer: B. n = 6 2/3
Hope this helps!
A spherical gas tank has a 10 foot diameter when filled with propane it provides 358,000,000 BTUs how many BTUs does 1 cubic foot of propane yield? Round to the nearest thousand
The number of BTUs does 1 cubic foot of propane yield is 684,000 BTUs/ft³ (in thousand)
Calculation of the number of BTUs:
Since we know that
The volume of the spherical tank is
[tex]V=4\div 3\pi r^3[/tex]
Now the volume of the tank will be:
[tex]V=4\div 3\times \pi \times (10/2)^3[/tex]
=523.59877756
= 523.6 ft³
Now the number should be
= [total amount] ÷ [total volume]
=358,000,000 ÷ 523.6
=683,728.0367 BTUs/ft³
= 684,000 BTUs/ft³ (in thousand)
Therefore, The number of BTUs does 1 cubic foot of propane yield is 684,000 BTUs/ft³ (in thousand)
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What is the name of the polygon if the sum of the measures of its interior angles is 1080°?
Answer:
Octagon
Step-by-step explanation:
It has 8 sides
During the first part of a trip, a canoeist travels 96 miles at a certain speed. the canoeist travels 15 miles on the second part of the trip at a speed 5 mph slower. the total time for the trip is 4 hrs. what was the speed on each part of the trip?
[tex]\text{x = }\dfrac{ -b \pm \sqrt{b^{2} - 4ac } }{2a} [/tex]
a = 4
b = - 131
c = 480
[tex]\text{x = }\dfrac{ -(-131) \pm \sqrt{(-131)^{2} - 4*4*480 } }{2*4} [/tex]Rovide an appropriate response.4) assume that adults have iq scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the wechsler test). find the iq score separating the top 14% from the others. 4)