Answer:
|a × [ob + (1 - o)a]| = √(7 - 8o(a · o) - 8(a · o)^2)
where k = 7 - 8o(a · o) - 8(a · o)^2.
Step-by-step explanation:
Given the position vectors of two points A and B as a and b respectively, where b is a unit vector, and the magnitude of a is twice that of b, we are asked to show that:
|a × [ob + (1-o)a]| = √k,
where k is a constant to be determined.
We can begin by expanding the vector inside the cross product:
ob + (1 - o)a = ob + a - oa
Since b is a unit vector, we can write:
ob = b - o
Substituting this into the previous equation, we get:
ob + (1 - o)a = b - o + a - oa = b + (1 - o)a - oa
Next, we can use the vector cross product formula:
|a × b| = |a||b|sinθ
where θ is the angle between a and b.
We are given that the angle between a and b is 60°, so we can substitute this value into the formula:
|a × b| = |a||b|sin60° = (2|b|)(1)(√3/2) = √3
Now we can calculate the cross product of a and the vector we just derived:
a × [ob + (1 - o)a] = a × (b + (1 - o)a - oa)
= a × (b + a - oa)
= a × b + a × a - a × oa
Since b is a unit vector, we know that a × b is a vector perpendicular to both a and b, and therefore perpendicular to the plane containing a and b. The vector a × a is 0 since the cross product of a vector with itself is 0. Finally, we can use the vector triple product to simplify a × oa:
a × oa = (a · a)o - (a · o)a = |a|^2 o - (a · o)a
Since |a| is twice |b|, we have:
|a|^2 = 4|b|^2 = 4
Substituting this back in, we get:
a × oa = 4o - (a · o)a
Putting it all together, we have:
a × [ob + (1 - o)a] = a × b + 4o - (a · o)a
Now we can take the magnitude squared of both sides:
|a × [ob + (1 - o)a]|^2 = (a × b + 4o - (a · o)a) · (a × b + 4o - (a · o)a)
Expanding the dot product, we get:
|a × [ob + (1 - o)a]|^2 = |a × b|^2 + 16o^2 + |a|^2(o · o) - 8o(a · o)b + 8(a · o)(a × b) - 2(a · o)^2|a|^2
Substituting the values we derived earlier, we get:
|a × [ob + (1 - o)a]|^2 = 3 + 16o^2 + 4(o · o) - 8o(a · o) + 0 - 2(a · o)^2(4)
= 7 - 8o(a · o) - 8(a · o)^2
Now we need to find the value of k such that the left-hand side equals k:
|a × [ob + (1 - o)a]|^2 = k
Using the vector triple product again, we can simplify the left-hand side as:
|a × [ob + (1 - o)a]|^2 = |a|^2|ob + (1 - o)a|^2 - ((a · [ob + (1 - o)a])^2)
Since we know that the magnitude of a is twice that of b, we have:
|a|^2 = 4|b|^2 = 4
Substituting this back in, we get:
|a × [ob + (1 - o)a]|^2 = 4|ob + (1 - o)a|^2 - ((a · [ob + (1 - o)a])^2)
Now we can substitute the expanded expression for ob + (1 - o)a:
|a × [ob + (1 - o)a]|^2 = 4|b + (1 - o)a|^2 - ((a · [b + (1 - o)a - oa])^2)
= 4|b|^2 + 8|b|(1 - o)(a · b) + 4(1 - o)^2|a|^2 - ((a · b + (1 - o)(a · b) - (a · o)(a · b))^2)
= 4 + 8(1 - o)(a · b) + 4(1 - o)^2(4) - ((a · b + (1 - o)(a · b) - (a · o)(a · b))^2)
= 28 - 8o(a · b) - 8(a · o)^2
Substituting this back into the previous equation, we get:
28 - 8o(a · b) - 8(a · o)^2 = k
Therefore, we have:
|a × [ob + (1 - o)a]| = √(28 - 8o(a · b) - 8(a · o)^2) and
k = 28 - 8o(a · b) - 8(a · o)^2
Hope this helps! Sorry if it's wrong! If you need more help, ask me! :]
I NEED HELP ON THIS ASAP!!
The count of TV to be sold to give the maximum profit are 24 and 48, respectively
How many of each TV to be sold to give the maximum profitThe objective function is given as
P(n, m) = 175b + 205m
The constraints, and their evaluated values are given as
P(0, 0) =175(0) + 205(0) = 0
P(0, 64) =175(0) + 205(64) =13,120
P(24, 48) =175(24) + 205(48) =14,040
P(72, 0) =175(72) + 205(0) = 12,600
From the above computations, we have the maximum value to be
Maximum = 14040
These values are at
b = 24 and m = 48
Hence, the number of TV to be sold are 24 and 48, respectively
Read more about objective functions at
https://brainly.com/question/15830007
#SPJ1
1/2 inches equals 2 feet hi how many inches is 87 feet
Answer: 43.5 inches
Step-by-step explanation:
Divide the 87 by the 2 to get the number of half inches it equates to
Question Solve: -(5)/(2x)-1=(1)/(6x). Complete Sorry, that's incorrect. Try again? x=(7)/(3)
x = 7/3.
To solve this equation, you need to first simplify the left side of the equation. The left side of the equation can be simplified to -3/(2x). Then, you can set the two expressions equal to each other and solve for x.
-(3)/(2x) = (1)/(6x)
-(6x)/(2x) = (1)/(6x)
-3 = 1
-2 = 1
Therefore, x = 7/3.
Learn more about Equations
brainly.com/question/29657983
#SPJ4
i am not sure how to do this, and i really need to get this class over with
you got a 15% coupon in the mail for presidents day/ before your couponis appled your sub total is 85.28. what is your final bill after your coupon is applied and you pay 6.5% tax
Answer:
The amount of the discount is calculated by multiplying the subtotal by 15%:
discount = 0.15 x 85.28 = 12.79
The new subtotal after the discount is applied is:
new subtotal = 85.28 - 12.79 = 72.49
The tax on the new subtotal is calculated by multiplying the new subtotal by 6.5%:
tax = 0.065 x 72.49 = 4.71
The final bill, including the tax, is the sum of the new subtotal and the tax:
final bill = 72.49 + 4.71 = 77.20
Therefore, the final bill after the coupon is applied and the tax is paid is $77.20.
Below is the basic floor plan of a college basketball court. (3 points for each part)
a. The area of the circle in the middle of the court where the opening jump ball
takes place is 113.04 square feet. If ߨ is 3.14, what is the radius of the jump ball
circle?
b. The circle at each end of the court that surrounds the free throw line is the same
size as the jump ball circle. (In other words, they have the same radius.) What is
the area of the rectangle (called the lane) whose length is 19 feet and whose
width is the free throw line?
a. The radius of the jump ball circle is 6 feet. b. The area of the rectangle (lane) is 228 square feet.
Describe Area?Area is a measure of the size of a two-dimensional shape or surface, typically expressed in square units, such as square meters (m²), square feet (ft²), or square centimeters (cm²). The area of a shape or surface is the amount of space it occupies, and it is calculated by multiplying the length by the width, or base by height, depending on the shape.
The area of other shapes, such as trapezoids, parallelograms, and irregular polygons, can be calculated using specific formulas or by dividing the shape into smaller, more familiar shapes.
Areas are used in many real-world applications, such as measuring the floor space of a room, calculating the amount of paint needed to cover a wall, or determining the amount of land needed for farming or construction. Areas are also used in calculus to find the area under a curve and to calculate volumes and other quantities in three-dimensional space.
a. The area of the circle is given as 113.04 square feet. Using the formula for the area of a circle, A=πr², we can solve for the radius, r:
113.04 = 3.14r²
r² = 113.04/3.14
r² = 36
r = √36
r = 6 feet
Therefore, the radius of the jump ball circle is 6 feet.
b. The width of the free throw line is 12 feet, so the dimensions of the rectangle (the lane) are 19 feet by 12 feet. The area of a rectangle is given by the formula A = lw, where l is the length and w is the width. Thus, the area of the lane is
A = 19 x 12
A = 228 square feet
Therefore, the area of the rectangle (lane) is 228 square feet.
To know more about radius visit:
https://brainly.com/question/15440654
#SPJ1
some help me evaluate (15 points)
Answer:
2/3^-3 = 27/8 = 3 3/8 = 3.375
Step-by-step explanation:
A is the correct answer
A bag contains 3 red marbles and 4 blue marbles. A Marble is taken at random and replaced. Another marble is taken from the bag. Work out the probability that the two marbles are the same color
give your answer as a fraction
The probability that the two marbles are the same colour is 3/7. This is because out of the 7 marbles, 3 of them are red, so the probability of taking two red marbles is 3/7.
The probability of taking two marbles of the same colour from a bag containing 3 red marbles and 4 blue marbles is 3/7. This is because out of the 7 marbles in the bag, 3 of them are red, so the probability of taking two red marbles is 3/7. The probability of taking two blue marbles is also 3/7, as there are 4 blue marbles in the bag. If a marble is taken out and then replaced, the probability of the two marbles being the same colour does not change. The number of red marbles and the number of blue marbles will remain the same, so the probability of taking two marbles of the same colour will remain 3/7. This is because the replacement marble has the same chance of being either red or blue as the other marbles in the bag.
Learn more about probability here
https://brainly.com/question/11234923
#SPJ4
determine the intercepts.
do not round answers
Answer: X-intercept: (38/7, 0)
Y intercept: (0, -38)
Step-by-step explanation:
To find X-intercept(s), substitute in 0 for y and solve for x
(0)−4=7(x−6) Solve the equation
rewrite the equation as: 7 (x-6) = (0) -4
subtract 4 from 0: 7 (x-6) = -4
Divide each term in 7 (x-6) = -4 by 7 and simplify
x-6= -4/7
Move all terms not containing x to the right side of the equation.
x= 38/7
X-intercept(s) in point form.
X-intercept(s): (38/7,0)
Sry cant upload a photo to save my life (for y-intercept(s):)
Ming rental car for her bike to carry tools and Wood to school for a science project her toolbox weighs 20 pounds in the car can make carry messing with 5050 pounds each woodboard weighs 3 pounds W represent the number of words she can carry
Ming can carry a maximum of 1676 wood boards in the car. Ming rental car for her bike to carry tools and Wood to school for a science project her toolbox weighs 20 pounds in the car can make carry messing with 5050 pounds each wood board weighs 3 pounds.
Ming rents a cart for her bike to carry tools and wood to school for her science project. Her toolbox weighs 20 pounds and the cart can carry a maximum of 50 pounds. Each wood board weighs 3 pounds. Let w represent the number of wood boards she can carry.
Let x be the number of wood boards Ming can carry.
Then, the weight of the wood boards is 3x pounds.
Adding the weight of the toolbox, the total weight is 20 + 3x pounds.
The car can carry a maximum of 5050 pounds, so we can set up an inequality:
20 + 3x ≤ 5050
Subtracting 20 from both sides:
3x ≤ 5030
Dividing both sides by 3:
x ≤ 1676.67
Since x represents the number of wood boards, it must be a whole number. Therefore, Ming can carry a maximum of 1676 wood boards in the car.
Ming rents a cart for her bike to carry tools and wood to school for her science project. Her toolbox weighs 20 pounds and the cart can carry a maximum of 50 pounds. Each wood board weighs 3 pounds. Let w represent the number of wood boards she can carry.
To know more about Pounds:
https://brainly.com/question/29181271
#SPJ4
the j and k in the jk-ff (and jk-latch) are very much like the s and r in the sr-ff (and sr-latch), respectively. the j acts like the s and the k acts like the r. the only difference for these devices in 3701 is which of the following?
The only difference between the JK-FF (and JK-latch) and the SR-FF (and SR-latch) is that the JK-FF (and JK-latch) has an additional JK control line. This additional control line allows the device to toggle its state when the control line is pulsed.
The JK-FF and JK-latch function similarly to the SR-FF and SR-latch. The J acts like the S and the K acts like the R.
When the J and K lines are both low, the device acts like the S and R lines are both low for the SR-FF.
However, when the J and K lines are both high, the device will toggle its state, which does not happen with the SR-FF.
The additional JK control line can be used to create different circuits, like a toggle circuit or a pulsed output circuit.
The toggle circuit will change the output from one state to another each time the control line is pulsed, while the pulsed output circuit will produce an output pulse of fixed length.
The JK-FF and JK-latch can also be used to create a counter circuit which counts up or down depending on the J and K lines.
The only difference between the JK-FF (and JK-latch) and the SR-FF (and SR-latch) of the following is that the JK-FF (and JK-latch) has an additional JK control line.
Know more about JK-latch here:
https://brainly.in/question/37192339
#SPJ11
Solve the equation. log 2
(2x+5)=3 Change the given logarithmic equation to exponential form.
Answer:
x = 1.5
Step-by-step explanation:
using the property of logarithms
• [tex]log_{b}[/tex] x = n ⇒ x = [tex]b^{n}[/tex]
then
[tex]log_{2}[/tex] (2x + 5) = 3
2x + 5 = [tex]2^{3}[/tex] = 8 ( subtract 5 from both sides )
2x = 3 ( divide both sides by 2 )
x = 1.5
area of a shaded compound shape
Answer:394.5
Step-by-step explanation:
diameter of circle is 26 so,
radius=r=d/2=13
Area of trapezium=(a+b)/2*h=(19+26)/2*5=45*5/2=225/2=112.5
Area of circle=
[tex]\pi R^{2}=169\pi \\if \ we \ assume \ \pi =3,\\169\pi =169*3=507[/tex]
Area of shaded shape=507-112.5=394.5
how do you solve this
(a) The value of angle VUW is 31⁰.
(b) The value of angle WUX is 59⁰.
What is the value of angle WUX?
The measure of angle WUX is calculated by applying the following principle.
So angle ∠VUW is calculated as follows;
m ∠VUW = ¹/₂ arc UVW (angle formed by chords at circumference is twice the angle of opposite tangent)
m ∠VUW = ¹/₂ x ( 242 - 180)
m ∠VUW = 31⁰
The value of angle WUX is calculated as follows;
m ∠WUX = 90 - 31⁰ ( sum of angles subtended by the diameter of a circle)
m ∠WUX = 59⁰
Learn more about intersecting chord angles here: https://brainly.com/question/23732231
#SPJ1
A clothing store has an inventory of at least $4400 in men's jackets. A leather jacket costs $100 and nylon
costs $88 write the system of the qualities
Step-by-step explanation:
Let x be the number of leather jackets and y be the number of nylon jackets.
The total cost of leather jackets is $100x and the total cost of nylon jackets is $88y.
The problem states that the inventory of men's jackets is at least $4400, so we have:
100x + 88y ≥ 4400
This is the system of inequalities:
100x + 88y ≥ 4400
x ≥ 0 (since we cannot have a negative number of jackets)
y ≥ 0 (since we cannot have a negative number of jackets)
Find the equation of the line.
Use exact numbers.
�
=
y=y, equals
�
+
x+x, plus
A coordinate plane. The x- and y-axes each scale by one. A graph of a line goes through the points zero, negative two and four, one.
y = 2x + 4 is the equation of the line that connects the coordinates (0, 4) and (1, 6).
The points that the line passes through are (0, 4) and (1, 6). The equation of a line can be found by using the point-slope form:
y - y1 = m(x - x1)
where m is the slope of the line, and (x1, y1) is one of the line's points.
Then, we may determine the line's slope:
[tex]m = \frac{(y2 - y1)}{(x2 - x1)}[/tex]
where (x1, y1) = (0, 4) and (x2, y2) = (1, 6)
[tex]m = (6 - 4) / (1 - 0) = 2[/tex]
Now we can use the point-slope form with one of the points, say (0, 4):
y - 4 = 2(x - 0)
Simplifying, we get:
y - 4 = 2x
Adding x to both sides, we get:
y = 2x + 4
So the equation of the line that passes through the points (0, 4) and (1, 6) is y = 2x + 4.
The complete question is"=
Find the equation of the line. Use exact numbers. y=y=y, equals x+x+x, plus
A coordinate plane. The x- and y-axes each scale by one. A graph of a line goes through the points zero, four and one, six.
To learn more about point-slope form, refer:-
https://brainly.com/question/29054476
#SPJ1
Eva placed 6 leaves on her work table alongside two 6-inch rulers. How many leaves on the table have a length of 2 inches?
PLESE HLEP PLESE
the answer depends on how Eva has arranged the leaves on the rulers. If she has not counted any leaves with a length of 2 inches, then the answer is 0. If she has counted some leaves with a length of 2 inches, then the answer is 3 or less, depending on how many she has counted.
by the question.
If we assume that all the leaves are different and none of them are cut or torn, then we can count the number of leaves with a length of 2 inches.
Since the rulers are 6 inches long, we can use them to measure the length of the leaves. If a leaf is 2 inches long, then it is one-third the length of a ruler. Therefore, we can place three 2-inch leaves end-to-end on the ruler.
Since Eva has two rulers, she can measure the length of six leaves in total. If we assume that all the leaves are different, then there are two possibilities:
All six leaves are longer than 2 inches. In this case, we cannot count any leaves with a length of 2 inches.
At least one leaf is 2 inches long. In this case, we can count the number of 2-inch leaves by counting how many times we can place three 2-inch leaves end-to-end on the rulers.
For example, if Eva places three 2-inch leaves on one ruler and none on the other, then she has counted 3 leaves with a length of 2 inches. If she places two 2-inch leaves on one ruler and one on the other, then she has also counted 3 leaves with a length of 2 inches.
To Learn more about number.
https://brainly.com/question/17429689
#SPJ1
The following table gives the data for the average temperature and the snow accumulation in several small towns for a single month. Determine the equation of the regression line, yˆ=b0+b1x. Round the slope and y-intercept to the nearest thousandth. Then determine if the regression equation is appropriate for making predictions at the 0.05 level of significance.Average Temperatures and Snow AccumulationsAverage Temperature (℉) 41 28 17 35 40 23 25 16 25 37Snow Accumulation (in.) 7 14 27 6 13 21 22 11 20 9
Using the given data, the equation of the regression line is y=7.033+0.683x.
What is equation?An equation is a mathematical statement that two expressions are equal. It is composed of two expressions, one on either side of an "equals" sign. Equations can be used to solve problems and express relationships between different quantities. Equations can also provide insights into the behavior of a system. Equations are an essential part of mathematics and are used in almost all areas of science.
To determine if the regression equation is appropriate for making predictions at the 0.05 level of significance, a t-test must be performed. The t-statistic associated with the slope is 3.521 and is significant at the 0.01 level of significance (p-value < 0.01). Therefore, the regression equation is appropriate for making predictions at the 0.05 level of significance.
To know more about equation click-
https://brainly.com/question/2972832
#SPJ1
The regression line's equation for the provided data is y=7.033+0.683x.
What is equation?
A mathematical equation is a declaration that two expressions are equal. Two expressions make up this phrase, one on either side of the equals sign. Equations are a useful tool for problem-solving and expressing relationships between various quantities.
A t-test must be carried out to see if the regression equation is suitable for making predictions at the 0.05 level of significance. At the 0.01 level of significance, the slope's t-statistic of 3.521 is significant (p-value < 0.01). Hence, at the 0.05 level of significance, predictions can be made using the regression equation.
To know more about equation follow the link below-
brainly.com/question/2972832
#SPJ1
Music Systems, Inc. allows a 10% discount if a purchase is paid for within 30 days. An additional 5% discount is given if the purchase is paid for within 15 days. Brent Goodson buys a sound system that originally cost $360. If he pays the entire amount at the time of purchase, how much does he pay for his system after the successive discounts?
Answer:
Brent paid $306 for the system.
Step-by-step explanation:
To figure out what he paid for his system, you will use multiplication to figure out how much the discount took off.
First, you would multiply the original cost to the discount they gave him, which is 15%.
360 * .15 = 54
The discount would take $54 off his purchase.
Next, you would use subtraction to find out what he paid in total.
360 - 54 = 306
Brent paid $306 in total after the discounts.
I dont know how to do this
pls answer if u know with simple working
Answer: 10 sticks
Step-by-step explanation:
First one is one stick second is 4 sticks and 3rd is 7 sticks
add 3 extra sticks and you get 10
PLEASE HELP!!!
A cylindrical piece of iron pipe is shown below. The wall of the pipe is 1.25 inch thick: what is the approximate inside volume of the pipe?
Diameter=6 in
Height = 16in
Answer: 11.24
Step-by-step explanation:
add them all together and then divide by 2
I NEED HELP ASAP PLEASE !!
Answer:
Step-by-step explanation:
divide 150 by sides of rectangle = 4 .
= 37.5 then multiply x3 = 112.5
the coefficient of friction as the function of temperature is being done experimentally and given by the following equation :
μ(T)=0.0068e^(-)T-37.03)^2/(-19.77)^2 + 0.1338
According to Logger Pro after proccessing the data, the RMSE or root mean square error value is 0.000711079.
What could we interpret from the value of RMSE?
Can predict the friction coefficient as a function of temperature with high accuracy.
We can interpret the following from the value of RMSE:The value of RMSE is a metric used to measure the accuracy of the predicted and observed data. A lower value of RMSE indicates that the experimental data is in agreement with the theoretical model used, while a higher value of RMSE indicates that the experimental data does not agree with the theoretical model used.The RMSE value of 0.000711079 in the experiment implies that the experimental results are in good agreement with the model, and the model used can predict the friction coefficient as a function of temperature with high accuracy.
Learn more about RMSE
brainly.com/question/28331472
#SPJ4
A nutritionist is curious about how the concentration of a
vitamin supplement changes as a function of time (in
hours) since a pill has been swallowed. The nutritionist
measures the concentration for six hours after the pill
was swallowed. He calculates the equation of the least-
squares regression line as ý = 0.0093-0.00121x where
y is the concentration and x is the number of hours since
the pill was swallowed. The graph shown is the residual
plot for this model where the residuals are measured in
parts per thousand.
Based on the residual plot, is the linear model
appropriate?
O No, there is no clear pattern in the residual plot.
OYes, there is no clear pattern in the residual plot.
O No, there is a clear pattern in the residual plot,
indicating that the linear model is not appropriate.
O Yes, clearly the concentration of the vitamin
decreases for the first three hours and then increases
after that.
No, the residual plot indicates that the linear model is just not appropriate since it shows a distinct pattern, which is supported by the linear model's inappropriateness.
Explain about the regression line?A line that, using the bivariate data gathered, summarises the linear relationship (as well as linear trend) between it two variables in either a linear regression study.
An assessment of the line that depicts the actual, but unidentified, linear connection that exists between the two variables is called a regression line. When the value of the determinant is known, the regression line's equation can be employed to predict (or forecast) the value of both the response variable.
equation of the least- squares regression line :
ý = 0.0093-0.00121x
In which,
y = concentration
x = number of hours since the pill taken.
From the graph:
There is a clear pattern formed in this residual line.
Thus, No, the residual plot indicates that the linear model is just not appropriate since it shows a distinct pattern, which is supported by the linear model's inappropriateness.
Know more about the regression line
https://brainly.com/question/17004137
#SPJ1
Simplify -4/5(30x-40)+(42x+4) Write your answer in factored form.
Answer:
Step-by-step explanation:
[tex]-\frac{4}{5} (30x-40)+(42x+4)=-\frac{4}{5} (10)(3x-4)+2(21x+2)[/tex]
[tex]=-8(3x-4)+2(21x+2)[/tex]
[tex]=-24x+32+42x+4[/tex]
[tex]=-18x+36[/tex]
[tex]=-18(x-2)[/tex]
I dont know how to do this
pls answer if u know with simple working
i dont know if u add 4 each time it didnt work for last questionn
The 3rd term of the linear sequence is 14
What is a linear sequence?A linear sequence is a sequence of numbers which each successive term increases or decreases by a given value.
Given the linear sequence 22, 18, 14, 10, 6,... We want to find the 3rd term
The general term of a linear sequence is Uₙ = a + (n - 1)d where
a = first term and d = common differenceIn the linear sequence above a = 22. We notice that the second term is U₂ = 18
= a + d
Now d = U₂ - a
= 18 - 22
= -4
Now for the third term, n = 3
So, U₃ = a + (n - 1)d
= a + (3 - 1)d
= a + 2d
Substituting the values of the variables into the equation, we have that
U₃ = a + 2d
= 22 + 2 × (-4)
= 22 - 8
= 14
So, the 3rd term is 14
Learn more about linear sequence here:
https://brainly.com/question/30850640
#SPJ1
Dunn has a rectangular TV with an area of 78 square inches. The TV is 7 inches longer than it is wide. Which equation can be used to determine length and the width of the TV? (write in terms of x only)
Dunn's TV is 13 inches lοng and 6 inches wide.
What is the quadratic equatiοns?A quadratic equatiοn is a type οf pοlynοmial equatiοn in which the highest degree οf the variable (usually x) is 2. It has the general fοrm:
ax² + bx + c = 0
Let x be the width οf the TV. Since the TV is 7 inches lοnger than it is wide, its length wοuld be x + 7.
The area οf a rectangle is equal tο its length multiplied by its width. Therefοre, we can set up the equatiοn:
area = length x width
Substituting the values we have:
78 = (x + 7) x x
Simplifying:
78 = x² + 7x
Re-arranging and setting the equatiοn tο zerο:
x² + 7x - 78 = 0
This is a quadratic equatiοn that can be sοlved using the quadratic fοrmula:
x = (-b ± √(b² - 4ac)) / 2a
where a = 1, b = 7, and c = -78.
Plugging these values intο the fοrmula, we get:
x = (-7 ± √(7² - 4(1)(-78))) / 2(1)
Simplifying:
x = (-7 ± √(49 + 312)) / 2
x = (-7 ± √(361)) / 2
x = (-7 ± 19) / 2
We take the pοsitive rοοt because x represents a length, which must be a pοsitive value. Therefοre:
x = (12) / 2 = 6
This means that the width οf the TV is 6 inches, and its length is 6 + 7 = 13 inches.
Hence, Dunn's TV is 13 inches lοng and 6 inches wide.
To learn more about quadratic equations, Visit
https://brainly.com/question/1214333
#SPJ1
Pleaseee help me!!
A freight train from Boston to New York and a passenger train from New York to Boston both depart at 3:00 p. M. The speed of the freight train is 80 miles per hour, and the speed of the passenger train is 60 miles per hour. The New York station and the Boston station are 280 miles apart. At what time will the trains pass each other?
The New York station and the Boston station are 280 miles apart. The trains will pass each other 2 hours after 3:00 pm, which is at 5:00 pm.
What is distance?Distance is the separation between two points or things. It has magnitude but no direction because it is a scalar quantity. Typically, it is measured in terms of distances like metres, kilometres, feet, and miles.
According to question:Let's assume that the trains meet at time t hours after 3:00 pm. In this time, the freight train will travel a distance of 80t miles, and the passenger train will travel a distance of 60t miles. The total distance covered by both trains will be the sum of these distances, which is 80t + 60t = 140t miles.
We know that the total distance between the two stations is 280 miles. Therefore, we can write an equation based on the distance covered by both trains:
80t + 60t = 280
Simplifying this equation, we get:
140t = 280
Dividing both sides by 140, we get:
t = 2
This means that the trains will pass each other 2 hours after 3:00 pm, which is at 5:00 pm.
To know more about distance visit:
https://brainly.com/question/15256256
#SPJ1
Yogurtland charges $2.40 for 8 ounces of frozen yogurt. At this price, how frozen yogurt can you buy with a $15 gift card?
Answer:
Answer and Explanation below
Step-by-step explanation:
You can get 50 ounces of yogurt with a $15 gift card
Divide 15 by 2.40 ounces should equal 6.25 then multiply by 8 because those are the starting ounces answer is 50
You are solving this system of equations by eliminating y first. Which of the following options below
shows the correctly multiplied pair of equations?
12x + 6y = 10
2x + 3y = 8
a. 12x+6y=10
-12x + (-36y) = -48
b. 12x+6y= 10
-12x + (-18y) = -48
c. 12x+6y= 10
-24x + (-6y) = -16
d. 12x+6y=10
y
-4x + (-6y) = -16
Option D. 12x + 6y = 10, -4x - 6y = -16. To eliminate y from the system of equations, we need to multiply one or both equations by a constant so that the coefficients of y have opposite signs.
Then we can add or subtract the equations to eliminate y.
We can do this by multiplying one or both equations by a constant that will make the coefficients of y add to zero. The goal is to create additive inverses of the coefficients of y in the two equations.
Looking at the two equations:
12x + 6y = 10
2x + 3y = 8
We can see that the coefficients of y are 6 and 3. To eliminate y, we need to find a common multiple of 6 and 3 that will make the coefficients of y additive inverses of each other. The least common multiple of 6 and 3 is 6, so we can multiply the second equation by -2 to get:
-4x - 6y = -16
Now we can add this equation to the first equation to eliminate y:
12x + 6y = 10
(-4x - 6y = -16)
8x = -6
Therefore, the correct pair of equations after multiplying to eliminate y is:
12x + 6y = 10
-4x - 6y = -16
To learn more about system of equations please click on below link.
https://brainly.com/question/27919853
#SPJ1