We have the equation:
[tex]49a^2-16=0[/tex]We can factorize this equation as:
[tex]\begin{gathered} 49a^2-16=0 \\ (7a)^2-4^2=0 \\ (7a-4)(7a+4)=0 \end{gathered}[/tex][tex]\begin{gathered} 7a-4=0\longrightarrow a_1=\frac{4}{7} \\ 7a+4=0\longrightarrow a_2=-\frac{4}{7} \end{gathered}[/tex]In this case, we have 2 rational solutions.
If the solution implies the square root of -1, then we would have 2 complex solutions.
If the solution implies a square root that does not have a rational solution, then we have 2 irrational solutions.
We can see it when we apply the quadratic formula:
[tex]x=-\frac{b}{2a}\pm\frac{\sqrt[]{b^2-4ac}}{2a}[/tex]The term with the square root defines what type of solution we have:
If b^2-4ac<0, then we have complex solutions.
If the square root of b^2-4ac does not have a rational solution (b^2-4ac is not a perfect square), then we have irrational solutions.
If b^2-4ac is a perfect square (its square root have a rational solution), we will have rational solutions.
sherry needs to borrow $6,200 to replace the air conditioner in her home. from her credit union, sherry obtains a 30 month loan with an annual simple interest rate of 5.75% use the formula I=Prt (time in years)
Simple Interest Equation (Principal + Interest)
A = P(1 + rt)
Where:
A = Total Accrued Amount (principal + interest)
P = Principal Amount
I = Interest Amount
r = Rate of Interest per year in decimal; r = R/100
R = Rate of Interest per year as a percent; R = r * 100
t = Time Period involved in months or years
Calculation:
First, converting R percent to r a decimal
r = R/100 = 5.75%/100 = 0.0575 per year.
Putting time into years for simplicity,
30 months / 12 months/year = 2.5 years.
Solving our equation:
A = 6200(1 + (0.0575 × 2.5)) = 7091.25
A = $7,091.25
The total amount accrued, principal plus interest, from simple interest on a principal of $6,200.00 at a rate of 5.75% per year for 2.5 years (30 months) is $7,091.25.
7. A right triangle has a hypotenuse of 13 and a leg of 8. What is the other leg? Show your work.
the other leg is √105
Explanation:Given:
Hypotenuse = 13
one of the legs of the triangle = 8
To find:
the other leg of the triangle
The triangle is right-angled. So, to get the third side, we will apply Pythagoras theorem:
Hypotenuse² = opposite² + adjacent²
let opposite = leg 1 = 8
adjacent = leg 2
Hypotenuse² = = leg1² + leg2²
[tex]\begin{gathered} 13^2=\text{ 8}^2\text{ + leg}_2^2 \\ \\ 169\text{ = 64 + leg}_2^2 \\ \\ 169\text{ - 64 = leg}_2^2 \\ \\ 105\text{ = leg}_2^2 \end{gathered}[/tex][tex]\begin{gathered} square\text{ root both sides:} \\ \sqrt{105}\text{ = leg}_2 \\ Can^{\prime}t\text{ be reduced an further inradical form} \\ \\ leg_2\text{ = }\sqrt{105}\text{ } \end{gathered}[/tex]Hence, the other leg is √105
What is the area of the portion of the triangle that lies outside of the circle but within the triangle? Provide the answer along with an explanation of how to calculate the answer.
Answer:
Explanation:
Given:
To determine the area of the portion of the triangle that lies outside of the circle but within the triangle, we find the areas of triangle and circle first:
For the triangle, we use the formula:
A=1/2bh
where:
b=base
h=height
We plug in what we know:
[tex]\begin{gathered} A=\frac{1}{2}bh \\ =\frac{1}{2}(20ft)(20ft) \\ =\frac{1}{2}(400ft^2) \\ \text{Calculate} \\ A=200ft^2 \end{gathered}[/tex]Next, we solve for the area of the circle using the given formula:
A=πr^2
where:
r=radius
So,
[tex]\begin{gathered} A=\pi r^2 \\ =\pi(6ft)^2 \\ \text{Calculate} \\ A=113.1ft^2 \end{gathered}[/tex]Then, to find the area of the portion of the triangle that lies outside of the circle but within the triangle:
Area of the portion = Area of the Triangle - Area of the Circle
We plug in what we know:
[tex]\begin{gathered} \text{ }=200ft^2-113.1ft^2 \\ \text{Area of the portion = }86.9ft^2 \end{gathered}[/tex]Therefore, the answer is 86.9 ft^2.
Perform the indicated operations and simplify the result so there are no quotients.
Given an expression:
[tex]\csc \theta(\sin \theta+\cos \theta)[/tex]We have to simplify the given expression.
[tex]\begin{gathered} \csc \theta(\sin \theta+\cos \theta)=\csc \theta\sin \theta+\csc \theta\cos \theta \\ =\frac{1}{\sin\theta}\cdot\sin \theta+\frac{1}{\sin\theta}\cdot\cos \theta \\ =1+\frac{\cos \theta}{\sin \theta} \\ =1+\cot \theta \end{gathered}[/tex]Thus, the answer is 1 + cot theta.
emily bought a pair of headphones on sale, if they usually cost 36.80, and emily purchased them at 40% how much did she pay? PLEASE GIVE ME A STEP BY STEP EXPLANATION PLEASE
Emily will pay $22.08 to buy the headphones after the discount of 40%.
Define percentage discountWhen a product is offered at a discount, the seller is reducing the product's price from what it was originally listed for. The percentage discount on the product is determined by the discount amount stated as a percentage of the original price.
Given data -
Cost of headphones - $36.80
Total price of product is considered as a 100%.
As Emily purchased them at 40%, she should be paying 60% of the amount.
Therefore, the amount she is paying to buy the headphones
= 60% of $36.80
= [tex]\frac{60}{100}[/tex] * $36.80
= 0.6 * $36.80
= $ 22.08
Emily will pay $22.08 to buy the headphones.
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Which expression is equivalent to 6c^2d^5/2c-2c^10/d^-2 ? A. 8c^11d^3B. 5c^13d^7C. 6c^11d^7D. 6c^12d^3
for each problem, identify the variables, write the equations, and solve
Let the 4-passenger cars be represented with F
Let the 6-passenger cars be represented with S
Rocket Coaster has 15 cars
So that F + S = 15 ----- Equation 1
Also, we were told that the total room for 72 passenger
so that 4F + 6S = 72 ----- Equation 2
Solving the two equations simultaneously using the substitution method,
Step 1
From equation 1:
Make F the subject of the formula
F = 15 - S ---- Equation 3
Step 2
substitute equation 3 into equation 2
4 (15 - S) + 6S = 72
Step 3
60 - 4S + 6S = 72
6S - 4S = 72 - 60
2S = 12
divide both sides by 2
S = 12/ 2
S= 6
Step 4
Substitute the value of S = 6 into equation 3
F = 15 - 6
F = 9
So the number of 4-passenger cars = 9
the number of 6-passenger cars = 6
can someone please help me find the valu of X
Two triangles than make the whole triangle are proportional then
[tex]15\longrightarrow30[/tex]and the upper side of the left triangle is x
we can calculate the lenght of upper triangle
[tex]36-15=21[/tex]then
[tex]21\longrightarrow x[/tex]we join the relations
[tex]\begin{gathered} 15\longrightarrow30 \\ 21\longrightarrow x \end{gathered}[/tex]we use cross-multiplication. The product of 21 by 30 dividing between 15 is equal to x
[tex]\begin{gathered} x=\frac{21\times30}{15} \\ \\ x=42 \end{gathered}[/tex]the value of x is 42 units
Find the perimeter of the figure below. Notice that one side length is not given.Assume that all intersecting sides meet at right angles.Be sure to include the correct unit in your answer.8 ftftft2ft9 ft151Х5?7 ft14 ftCheck2020 MCG Edus
We are asked to find the perimeter of the given figure.
Recall that the perimeter is basically the sum of all the sides in the figure.
But the length of one side is missing that can easily be found.
Let me draw the figure to illustrate the idea.
Step 1: Find the length of the missing side
As you can see in the above figure,
We found the length of the missing side by subtracting the length of 9 ft from the 14 ft, which will give us the length of the missing side drawn in red color.
Step 2: Find the parameter of the figure
Now we can proceed to find the perimeter of the figure.
Add up the lengths of each side
[tex]\begin{gathered} Perimeter=5ft+15ft+14ft+7ft+9ft+8ft \\ Perimeter=58ft \end{gathered}[/tex]Therefore,
The length of the missing side is 5 ft
The perimeter of the figure is 58 ft
Note:
The unit of the perimeter of the rectangle is ft
Whereas the unit of the area of the rectangle is ft^2
4) A quart of liquid has a mass of 2.70 kilograms. How many quarts will take to weight 100 pounds?
We will have the following:
1)
[tex]10\operatorname{km}\cdot\frac{0.621mi}{1\operatorname{km}}=6.21mi[/tex]2)
[tex]\frac{14}{2}\frac{bales}{\text{week}}\cdot\frac{52\text{week}}{1\text{year}}=364\frac{bales}{\text{wek}}[/tex]3)
[tex]\frac{85.4\text{yard}}{5\min}\cdot\frac{1\text{min}}{60s}\cdot\frac{16\text{inch}}{1\text{yard}}\cdot\frac{2.54\operatorname{cm}}{1\text{inch}}\cdot\frac{1m}{100\operatorname{cm}}\approx0.1156885333\ldots\frac{m}{s}[/tex]4)
[tex]100lb\cdot\frac{1\operatorname{kg}}{2.2lb}\cdot\frac{1\text{quart}}{2.70\operatorname{kg}}=\frac{5000}{297}\text{quarts}\approx16.83501684\ldots quarts[/tex]***Screenshot***
I sent the attachment cuz I rather not type :3
If two matrices are equal, then each of its elements must be equal.
If:
[tex]\begin{bmatrix}{a+3} & {4} \\ {6} & {b-1}\end{bmatrix}=\begin{bmatrix}{-3} & {4} \\ {6} & {2}\end{bmatrix}[/tex]This means that:
[tex]\begin{gathered} a+3=-3 \\ 4=4 \\ 6=6 \\ b-1=2 \end{gathered}[/tex]Isolate a and b from their respective equations to find their value:
[tex]\begin{gathered} a+3=-3 \\ \Rightarrow a=-3-3 \\ \therefore a=-6 \end{gathered}[/tex][tex]\begin{gathered} b-1=2 \\ \Rightarrow b=2+1 \\ \therefore b=3 \end{gathered}[/tex]Therefore, the value of a is -6 and the value of b is 3.
An object is thrown with an initial velocity of 10 m/s off a cliff that is 660 m high. Use the formula s=4.9t^2 + v0t.a.How long does it take for the object to hit the ground?b. How far will it fall in 3 seconds?
hello
to solve this question, we proceed to apply the formula given
initial velocity v0 =
[tex]\begin{gathered} v_0=10\text{ m/s} \\ s=660m \end{gathered}[/tex]next we proceed to substitute the values into the equation
[tex]\begin{gathered} s=4.9t^2+v_0t \\ 600=4.9t^2+10t \\ 4.9t^2+10t-600=0 \end{gathered}[/tex]we'll proceed to solve this quadratic equation to find the time it took the ball to hit the ground
there are several methods to solve a quadratic equation and for the purpose of this session, i'll make use of quadratic formula
[tex]t=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]in our equation, we have
[tex]\begin{gathered} a=4.9 \\ b=10 \\ c=-600 \end{gathered}[/tex]let's substitute the values into the equation
[tex]\begin{gathered} t=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ t=\frac{-10\pm\sqrt[]{10^2-4\times(4.9\times-600)}}{2\times4.9} \\ t=\frac{-10\pm\sqrt[]{100+11760}}{9} \\ t=\frac{-10\pm\sqrt[]{11860}}{9} \\ t=\frac{-10\pm108.9}{9} \\ t=\frac{98.9}{9}\text{ or t}=\frac{-118.9}{9} \\ t=10.99\text{ ot t}=-13.21 \end{gathered}[/tex]but since time can't have a negative value, the answer is approximately 10.99.
the time taken for the object to hit the ground is approximately 10.99s
b.
how far will it fall in 3s
to find the distance the object travelled in 3s, let's substitute the value of time in the original equation
[tex]\begin{gathered} s=4.9t^2+v_0t \\ t=3 \\ v_0=10\text{ m/s} \\ s=4.9(3)^2+10\times3 \\ s=4.9\times9+30 \\ s=44.1+30 \\ s=74.1m \end{gathered}[/tex]in 3s, the ball travelled a distance of 74.1m
Sam has $4.30 in quarters and nickels. He has 26 coins in all. The following system of equations can be used to represent this situation. 0.25q +0.05n = 4.30 q+n=26 What does the solution to the system (q, n) represent? (4 pts.) A.There is not enough information to determine. B.The value of the quarters and nickels Sam has C.The number of quarters and the number of nickels Sam has. D.The total amount of money Sam has.
ANSWER
C.The number of quarters and the number of nickels Sam has.
EXPLANATION
We have that the system of equations representing the situation is:
0.25q + 0.05n = 4.30
q + n = 26
The solution to the system of equations (q, n) will represent the number of quarters and nickels that Sam has.
This is because the sum of q and n is 26 and we are told that Sam has 26 coins.
Therefore, we can conclude that (q, n) represents the number of quarters and nickels that Sam has.
Write the congruence statements represented by the markers in each diagram
The congruence statements
PS= QR<PSQ= <QRP
<UTV= <VWX<TUV= <XWV
What is Congruence?If all three corresponding sides and all three corresponding angles are equal in size, two triangles are said to be congruent. Slide, twist, flip, and turn these triangles to create an identical appearance.
Given:
From the Figure
PS= QR
<PSQ= <QRP
and, from another figure
<UTV= <VWX
<TUV= <XWV
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If Matthew rolls a number cube 90 times, how many times can he expect it 2 poilto land on an odd number?
Solution
The total face of a cube is 6
Possible odd 1,3,5
[tex]Probability\text{ =}\frac{Possible\text{ outcome}}{Total\text{ outcome}}[/tex]The cube is numbered one through six, and the odd numbers are therefore 1,3,5. So the probability of rolling an odd number in any given single roll is
[tex]\frac{3}{6}=\frac{1}{2}[/tex]Multiplying this probability by the number of times she rolls the cube gives
[tex]\frac{1}{2}\times90=45[/tex]The final answer
[tex]45[/tex]Given 8 x = 2 y - z, solve for y if x 2 and z = 2.
Answer
y = 9
Explanation
The question wants usto solve for y, given that
8x = 2y - z
if x = 2 and z = 2
To solve this, we just substitute the given values for x and z into the given expression to get y.
8x = 2y - z
8(2) = 2y - 2
16 = 2y - 2
We can rewrite this and have
2y - 2 = 16
Add 2 to both sides
2y - 2 + 2 = 16 + 2
2y = 18
Divide both sides by 2
(2y/2) = (18/2)
y = 9
Hope this Helps!!!
2. Which of the following statements istrue? (1 pt)a) The longest side in a right triangle iscalled the hypotenuse.b) Pythagorean Theorem works for alltriangles.c) The Pythagorean Theorem is used to findmissing angles.
Option A is correct
Explanation:Pythagorean Theorem only work for right angle triangles.
Hence, the option Pythagorean Theorem works for all triangles is wrong
The Pythagorean Theorem is used to find missing sides of right angle triangles.
The option The Pythagorean Theorem is used to find missing angles is wrong
The longest side in a right triangle is called the hypotenuse.
This is correct
Option A is correct
Rewrite all the equation using the inverse operation.I WILL SEND PICTURES OF PROBLEM
Explanation
Step 1
[tex]\begin{gathered} a+15=32 \\ \text{subtract 15 in both sides} \\ a+15-15=32-15 \\ a=17 \end{gathered}[/tex]Step 2
[tex]\begin{gathered} b-12=75 \\ \text{add 12 in both sides} \\ b-12+12=75+12 \\ b=87 \end{gathered}[/tex]Step 3
[tex]\begin{gathered} 9x=90 \\ we\text{ n}eed\text{ isolate x, so divide both sides by 9} \\ \frac{9x}{9}=\frac{90}{9} \\ x=10 \end{gathered}[/tex]Step 4
[tex]\begin{gathered} \frac{x}{6}=7 \\ \text{Multiply both sides by 6} \\ \frac{x}{6}\cdot6=7\cdot6 \\ x=42 \end{gathered}[/tex]I hope this helps you
Time 13.5 Time 16 Time 12.6 Time 15.2 Time 12.8 Time 11.8 Time 17.2 Time 12.1. what is the difference between Sarah's time and the mean time of the runners. I know that you divide eight by the sum when you add up all the numbers together from smallest to largest and could you answer this for me l really don't in a word problem when to multiply, add, divide,or, subtract could you ex plain or tell what the key words are that let a student know to add, multiply ,divide, subtract the key words please.
The mean of a data set is equal to the addition of the values of the data divided by the total number of values in the set.
In our problem:
[tex]\begin{gathered} \operatorname{mean}=\frac{13.5+16+12.6+15.2+12.8+11.8+17.2+12.1}{8}=\frac{111.2}{8} \\ \Rightarrow\operatorname{mean}=13.9 \end{gathered}[/tex]Although you already calculated that, as I understand from what you wrote in the question tab.
The median of a data set can be found by ordering the values in the data set from least to greatest and then taking the middle value. In our problem:
[tex]\begin{gathered} 11.8,12.1,12.6,12.8,13.5,15.2,16,17.2 \\ \Rightarrow\operatorname{median}=\frac{12.8+13.5}{2}=13.15 \\ \Rightarrow\operatorname{median}=13.15 \end{gathered}[/tex]How to divide 111.2 by 8:
The difference between Sarah's time and the mean time of the runners is:
[tex]\operatorname{mean}-\text{Sarah}=13.9-12.1=1.8[/tex]The answer we are looking for is 1.8
In a survey, 221people said theyhave cable TV. Thisrepresents 65% of thepeople surveyed.How many peoplewere surveyed?
1) Gathering the data
221 people = 65%
2) To find the 100% people surveyed let's set a direct proportion. And then cross multiply to have an equation:
% people
65-----------221
100-------- y
65y = 22100 Dividing both sides by 65
y =340
So, 340 people were surveyed.
Designing Doorways for Rich and RizPON D5512ProblemRich has plenty of space abovenstead in this may Rizis quite a bitWat het oude budhat would give the same amount ofhadicom proportionatasRepen555sony net 50
Step 1
Calculate the difference in Rich height and the height of the doorway
6' 67/100'' - 5 5 3/4''
= 1' 19/100''
Then add the difference to Riz height
1' 19/100'' + 6' 8 1/2''
= 7' 9/10''
The height of the doorway that would give Riz enough space is 7' 9/10''
find a decimal that is equal to each fraction. round to the nearest thousandth if necessary 271/100
Here, we want to find the decimal equal to the given fraction
To do this, we look at the number which is the denominator
The number here is 100
What this mean is that we are going to shift the decimal point two times (due to 2 zeros; if 1000, 3 times)
The decimal point is not visible on the numerator which means it is at the back of the last number 1 but it is not necessary to write it
By virtue of the movement, the decimal point will be moved two times, which will make it land at the back of the first number 2
So, we have the expression as;
[tex]\frac{271}{100}\text{ = 2.71}[/tex]Solve the following linear equation using equivalent equations to isolate the variable. Express youranswer as an integer, as a simplified fraction, or as a decimal number rounded to two places.2345 2----U43НЕ KeyAnswerHow to enter your answer (Opens in new window)Keyboard ShoU=
Given the equation
[tex]\frac{-3}{4}-\frac{2}{3}=\frac{5}{4}u-\frac{2}{3}u[/tex]We add and subtract similar terms:
[tex]-\frac{17}{12}=\frac{7}{12}u[/tex]Multiply both sides by 12:
[tex]\begin{gathered} -\frac{17}{12}\cdot12=12\cdot\frac{7}{12}u \\ -17=7u \end{gathered}[/tex]Divide both sides by 7:
[tex]\begin{gathered} -\frac{17}{7}=\frac{7u}{7} \\ u=-\frac{17}{7} \end{gathered}[/tex]Answer:
[tex]u=-\frac{17}{7}[/tex]c) blank + 48 minutes= 16:00
15:12
Explanation
to find the missing value we need to solve the equation
[tex]blank+48minutes=16\colon00[/tex]so, to isolate the blank subtract 48 minutes in both sides
[tex]\begin{gathered} blank+48minutes-48minutes=16\colon00-48minutes \\ blank=16\colon00-48minutes \end{gathered}[/tex]To subtract time,subtract minutes then subtract the hours, since we can not have negative minutes , add 60to minutes and subtract 1 from the hours , so
[tex]16\colon00\text{ =15 hours +60 minutes}[/tex]replace and do the subtraction
[tex]\begin{gathered} blank=16\colon00-48minutes \\ blank=15\colon00-48minutes+60\text{minutes} \\ blank=15\colon12 \end{gathered}[/tex]therefore, the answer is
15:12
I hope this helps you
Evaluate / Solve b3 + ac – bif a = 4, b=5, and c = 10.A.170B.165C.160D.125
Answer
Option C is correct.
The answer is 160.
Explanation
We are told to evaluate or solve
b³ + ac - b
if a = 4, b = 5 and c = 10
So, to solve, we will just substitute the given values into the expression to be solved
b³ + ac - b
= (5)³ + (4 × 10) - 5
= 125 + 40 - 5
= 160
Hope this Helps!!!
In ∆JKL, j=7.9inches, k=2 inches and l =9.8. find the measure of
By cosine rule,
[tex]\cos K=\frac{j^2+l^2-k^2}{2jl}[/tex]Where j = 7.9 inch, k = 2 inch and l = 9.8
[tex]\cos K=\frac{7.9^2+9.8^2-2^2}{2\times7.9\times9.8}=\frac{154.45}{154.84}=0.99748[/tex][tex]\begin{gathered} \cos K=0.99748 \\ K=\cos ^{-1}0.99748=4.067\approx4^o \end{gathered}[/tex]Solution: The measure of angle K is 4 degrees
What is the answer for v/3-1=3
Given the initial expression
[tex]\frac{v}{3}-1=3[/tex]Solve for v as shown below
[tex]\begin{gathered} \frac{v}{3}-1=3 \\ \Rightarrow\frac{v}{3}-1+1=3+1 \\ \Rightarrow\frac{v}{3}=4 \\ \Rightarrow3\cdot\frac{v}{3}=3\cdot4 \\ \Rightarrow v=12 \end{gathered}[/tex]The answer is v=12
Find the vertex of the quadratic function below. Write your answer in fractional form if necessary. Do not use decimals.
Given quadratic expression is
[tex]f(x)=-3x^2-4x+7[/tex]Comparing with the general quadratic function
[tex]f(x)=ax^2+bx+c[/tex]We have,
[tex]a=-3,b=-4,c=7[/tex]So, the x coordinate of the vertex is
[tex]\begin{gathered} -\frac{b}{2a}=-\frac{-4}{-6} \\ =-\frac{2}{3} \end{gathered}[/tex]Now, putting the x coordinate in the equation of the given function,we have,
[tex]\begin{gathered} f(-\frac{2}{3})=-3(\frac{4}{9})-4(-\frac{2}{3})+7 \\ =-\frac{12}{9}+\frac{8}{3}+7 \\ =\frac{75}{9} \end{gathered}[/tex]So, the vertex of th egiven quadratic function is at
[tex](-\frac{2}{3},\frac{75}{9})[/tex]Can someone help me with this
The surface area of trapezoidal is 125 cm² that is the popcorn box has a surface area 125 cm².
Given that,
In the picture we have a popcorn with dimensions of trapezoidal.
We have to find what is the surface area of the trapezoidal.
We know that,
The surface area of trapezoidal is 1/2(b₁+b₂)×h
Here,
b₁=5 cm
b₂= 5cm
h= 25 cm
The surface area of trapezoidal= 1/2(b₁+b₂)×h
The surface area of trapezoidal= 1/2(5+5)×25
The surface area of trapezoidal= 1/2(10)×25
The surface area of trapezoidal= 5×25
The surface area of trapezoidal= 125
Therefore, The surface area of trapezoidal is 125 cm² that is the popcorn box has a surface area 125 cm².
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Convert: 3120 millimeters =centimeters
Given 1 centimeter = 10 milimeters.
To convert a number in millimeters into centimeters, divide the number by 10.
[tex]\begin{gathered} 3120millimeters=\frac{3120}{10}centimeters \\ =312centimeters \end{gathered}[/tex]Answer: 312 centimeters.