The diagonal of a square is x units. What is the area of the square in terms of x?
square units
square units
2x square units
square units
Answers
Let
b------> the length side of a square
we know that
the area of a square is equal to
[tex]A=b^{2}[/tex]
Find the length side of the diagonal applying the Pythagoras Theorem
[tex]d^{2}=b^{2}+b^{2}[/tex]
[tex]d^{2}=2b^{2}[/tex]
[tex]d=b\sqrt{2}\ units[/tex]
Remember that
[tex]d=x\ units[/tex] -----> given problem
substitute
[tex]d=b\sqrt{2}\ units[/tex]
[tex]x=b\sqrt{2}\ units[/tex]
Solve for b
[tex]b=\frac{x\sqrt{2}}{2}\ units[/tex]
Substitute in the formula of area
[tex]A=(\frac{x\sqrt{2}}{2})^{2}[/tex]
[tex]A=\frac{x^{2}}{2}\ units^{2}[/tex]
therefore
the answer is
[tex]\frac{x^{2}}{2}\ units^{2}[/tex]
The area of the square in terms of x is [tex]\( \frac{x^2}{2} \),[/tex] obtained by using the Pythagorean theorem and the formula for the area of a square.
To find the area of the square in terms of [tex]\(x\),[/tex] we need to first understand the relationship between the diagonal and the side length of a square.
In a square, each diagonal divides the square into two congruent right triangles. Using the Pythagorean theorem, we can find the relationship between the diagonal [tex](\(x\))[/tex] and the side length of the square [tex](\(s\)).[/tex]
The Pythagorean theorem states:
[tex]\[ \text{Hypotenuse}^2 = \text{Adjacent side}^2 + \text{Opposite side}^2 \][/tex]
For a square:
[tex]\[ x^2 = s^2 + s^2 \][/tex]
[tex]\[ x^2 = 2s^2 \][/tex]
Now, we can solve for [tex]\(s\)[/tex] in terms of [tex]\(x\):[/tex]
[tex]\[ s^2 = \frac{x^2}{2} \][/tex]
[tex]\[ s = \sqrt{\frac{x^2}{2}} \][/tex]
[tex]\[ s = \frac{x}{\sqrt{2}} \][/tex]
The area [tex](\(A\))[/tex] of a square is given by:
[tex]\[ A = s^2 \][/tex]
Substituting the expression for s in terms of x :
[tex]\[ A = \left(\frac{x}{\sqrt{2}}\right)^2 \][/tex]
[tex]\[ A = \frac{x^2}{2} \][/tex]
Therefore, the area of the square in terms of x is [tex]\( \frac{x^2}{2} \).[/tex]
The question probable may be:
The diagonal of a square is x units. What is the area of the square in terms of x?
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Temperature rose 9°f in 3 hours if the starting wad -5°f what was the final temperature
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dodecagon with a perimeter of 108 cm ...?
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Length of each side = 108/12 = 9 cm.
Length of the apothem = (9/2)/tan 15 = 9/2tan 15 = 16.79 cm
Area of each triangle = 9/2(16.79) = 75.57 cm^2
Therefore, area of the dodecagon = 12(75.57) = 906.9 cm^2
a 12 sided regular polygon is composed of 24 right triangles with one side
half the length of a side = 4.5 cm
the other side is 4.5 / tan (360/24) = 4.5 / tan(15)
A = 24 (1/2) 4.5^2 / tan(15) = 906.9 cm^2
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Which of the following is not an example of using the distributive property to multiply 12 times 22?
a. 12(20 + 2)
b. 22(10 + 2)
c. 10(2 · 22)
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It doesn't have the correct format and is mixed up.
3^x : write in terms of base e and simplify! ...?
Answers
3x=e^ln3^x
=e^xln3
Let
[tex]y = {3}^{x} [/tex]
Writing the above exponential equation as a logarithmic equation, we obtain;
[tex] log_{3}(y) = x[/tex]
We can apply the change of base formula to obtain,
[tex] \frac{ log_{e}(y) }{ log_{e}(3) } = x[/tex]
Or
[tex] \frac{ ln(y) }{ln(3) } = x[/tex]
We can cross multiply to obtain;
[tex] ln(y) = x ln(3) [/tex]
Taking logarithm of both sides to base e, we obtain;
[tex] {e}^{ ln(y) } = {e}^{x ln(3) } [/tex]
This implies that,
[tex]y= {e}^{x ln(3) } [/tex]
But we know that,
[tex]y = {3}^{x} [/tex]
Hence
[tex] {3}^{x} = {e}^{x ln(3)} [/tex]
What are the 2 types of unbiased samples?
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If M(-8, 12) is the midpoint of segment PQ, and P(8, 0), find the coordinates of Q.
Answers
Q(x2,y2)=(a,b)
M(x,y)=(-8,12)
x1+x2/2=-8;y1+y2/2=12
8+x2=-16;0+y2=24
x2=-24;y2=24
Q(x2,y2)=(-24,24)
"Find the value of the derivative (if it exists) at each indicated extremum. (If an answer does not exist, enter DNE.)
f (x) = cos(πx/2)
Answers
Answer: The derivate is f'(x) = -sin(πx/2)*(π/2)
Step-by-step explanation:
We want the value of the derivate at each point marked:
As you may know, the derivate of cos(x) is equal to -sin(x)
And in our case:
f(x) = cos(πx/2)
f'(x) = -sin(πx/2)*(π/2)
Where the rule used is that the derivate of f(g(x)) is equal to f'(g(x))*g'(x)
Now, as you may see, in both the marked points the curve changes of direction, so the tangent line in that point must be zero, letsinput the values of x in the derivate and see if this is true:
f'(0) = -sin(π*0/2)*(π/2) = -sin(0)*(π/2) = 0
f'(2) = -sin(π*2/2)*(π/2) = -sin(π)*(π/2) = 0
So this makes sense.
If you subtract the death rate from the birth rate of a population, you will calculate what?
A. the arithmetic growth rate.
B. the expansion rate.
C. the carrying capacity.
D. the intrinsic growth rate.
Answers
If you subtract the death rate from the birth rate of a population, you will calculate the carrying capacity.
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Subtracting the death rate from the birth rate of a population calculates its intrinsic growth rate, expressed as dN = (b - d)N, where 'r' is the intrinsic rate of increase.
When you subtract the death rate (D) from the birth rate (B) of a population, you calculate the intrinsic growth rate of the population. The birth rate is typically expressed on a per capita basis, shown as B = bN, where 'b' represents the per capita birth rate and 'N' is the number of individuals in the population. Similarly, the death rate is expressed as D = dN, where 'd' is the per capita death rate. To find the population growth rate at a specific point in time, differential calculus is used to obtain an instant-specific measurement:
dN = (b - d)N
This formula simplifies the relationship between the birth and death rates, where 'b - d' is substituted with 'r', representing the intrinsic rate of increase. Therefore, the correct answer to the student's question is D. the intrinsic growth rate.
When you multiply or divide an inequality by a ______ number, you must flip the inequality symbol.
Answers
A box has a width 9 inches less than its height and a length 7 inches more than its height.
a. If h is the height of the box, write a polynomial that represents the volume of the box.
b. Use this polynomial to calculate the volume for h = 17 inches.
Answers
l=h+7
w=h-9
a.
volume is height x width x length
v=h*(h+7)*(h-9)
now we have to multiply these values
h*(h+7)=h^2+7h
(h^2+7h)*(h-9)=h^3-9h^2+7h^2-63h=h^3-2h^2-63h
b.
now we plug in h=17
v=17^3 -2(17)^2 -63(17) = which comes out to 2686
note: **i suggest going through these steps on your own mostly because it's late and i'm tired and that seems like an awfully large number so you may want to check it
If 6 is 24% of a number, what is 40% of the same number?
(A) 8
(B) 10
(C) 15
(D) 20
Answers
To find 40% of the number where 6 is 24%, we first determine the whole number by dividing 6 by 0.24 to get 25. Then we calculate 40% of 25, which is 10.
If we know that 6 is 24% of a number, we can find the whole number by dividing 6 by 0.24. Once we have the full number, we can calculate 40% of it. Let's do the calculation step by step.
Calculate the whole number using the given percentage: Whole number = 6 / 0.24.
Perform the division to find the whole number: Whole number = 25.
Calculate 40% of the whole number: 40% of 25 = 0.40 * 25.
Perform the multiplication to find 40% of the number: 40% of 25 = 10.
Therefore, 40% of the same number is 10.
can u show work 39472/78 ...?
Answers
78 x 5 = 390
Subtracting, then bringing down 7
47 is smaller than 78 so
78 x 0 = 0
Bringing down 2, we get 472
78 x 6 = 468
Subtracting, the remainder is 4
The answer is 506 with a remainder of 4.
find the missing angle of the triangle.
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Compare 0.409 and 0.490 with >,<,=
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hope this helps you
your weekly wage is $400 what is your annual?
Answers
$400 x 52 = $28000
For a $1,000 loan at a rate of 16 3/4 % for 5 years, the simple interest is a. $83.75
b. $167.50
c. $500.00
d. $837.50
Answers
solve for x. round to nearest tenth if needed.
1. 7r-7=2r+18
a.r=-5
b.r=5
c.r=2.2
d.r=1.2
2. 2x+12=18-x
a.x=3
b.x=10
c.x=6
d.x=2
3. 8x-3=15x+18
a.x=-3
b.x=3
c.x=2.1
d.x=0.9
4. 6y-6=4y+16
a.y=2.2
b.y=-2.2
c.y=11
d.y=5
5. 3(x-4)=5(x+2)
a.x=11
b.x=-11
c.x=1
d.x=-1
Answers
-----------------
2x+x=18-12→3x=6→x=2
---------------
8x-15x=18+3→-7x=21→x=-3
----------------
6y-4y=16+6→2y=22→y=11
--------------
3x-12=5x+10→3x-5x=10+12→-2x=22→x=-11
On solving the linear equations given, we get the values as 5, 2, -3, 11 and -11 for the equations 7r - 7 = 2r + 18, 2x + 12 = 18 - x, 8x - 3 = 15x + 18, 6y - 6 = 4y + 16 and 3(x - 4) = 5(x + 2) respectively.
What is Linear Equation?Linear equations are equations which includes one or more variables and constants with the highest degree of the variable being 1.
1. 7r - 7 = 2r + 18
Taking the variables on one side and constants on the other side,
7r - 2r = 18 + 7
5r = 25
r = 5
2. 2x + 12 = 18 - x
2x + x = 18 - 12
3x = 6
x = 2
3. 8x - 3 = 15x + 18
8x - 15x = 18 + 3
-7x = 21
x = -3
4. 6y - 6 = 4y + 16
6y - 4y = 16 + 6
2y = 22
y = 11
5. 3(x - 4) = 5(x + 2)
Multiplying each term inside the bracket by the term outside,
3x - 12 = 5x + 10
3x - 5x = 10 + 12
-2x = 22
x = -11
Hence we got the values of the unknowns as 5, 2, -3, 11 and -11 for the equations 7r - 7 = 2r + 18, 2x + 12 = 18 - x, 8x - 3 = 15x + 18, 6y - 6 = 4y + 16 and 3(x - 4) = 5(x + 2) respectively.
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Solve the inequality.
p – 9 < –9
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p-9 < -9
p-9+9 < -9+9
p <0
name all opisties of -12
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The distance a spring will stretch varies directly with how much weight is attached to the spring. if a spring stretches 9 inches with 100 pounds attached, how far will it stretch with 90 pounds attached? round to the nearest tenth of an inch.
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Cross multiply to get 810 = 100x
Divide both sides by 100 to get x = 81 inches
A jug of juice holds 8 cups. convert 8 cups to pints
Answers
8 cups convert to pints = 4 pints
Further explanation7 main quantities have been determined based on international standards, namely:
1. Length, meters (m)2. Time, second (s)3. Mass, kilograms (kg)4. Temperature, kelvin (K)5. Light intensity, candela (cd)6. Electric current, ampere (A)7. Amount of substance, mol (m)Volume is a derivative quantity derived from the length
The unit of volume can be expressed in liters or milliliters or cubic meters
The conversion is
1 cc = 1 cm³
1 dm = 1 Liter
1 mL = 1 cm³
1 L = 1 dm³
The other common units for US standart volume are :
Fluid OuncesCupsPintsQuartsGallonsThe conversion for all those units :
1 gallon = 4 quarts = 8 pints = 16 cups = 128 fluid ounces
The conversion for pints and cups :
[tex]\large{\boxed{1\:pints=2\:cups}}}[/tex]
A jug of juice holds 8 cups
so 8 cups = 8 : 2 pints
8 cups = 4 pints
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f (x)+13=7x
x=f (x)+13/7
f^-1 (x)=x+13/7
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picture added.
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Because a negative square root is an i
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3^2+4^2=c^2
9+16=c^2
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sqr 25=sqr c^2
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Write a proportion that can be used to find the cost of 10 notebooks if 3 notebooks cost $1.98.
Answers
For 3 notebooks you pay $1.98 so the ratio is 1:0.66.
For 10 notebooks, then we could scale the ratio and it will become 10:6.60. So they will cost you $6.60.
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What is the inverse of f(x)=-3x^3 6?
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the midpoint of a segment is (2,-5), and one of the endpoints is (3,6). where is the other endpoint?
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(3, 6) is (2+1, -5+11) is an end point
therefor the other end point should be (2-1, -5-11) = (1, -16)
You could visualise this better if you draw it
Hope this helps, I don't really know a better way to explain it, sorry =P
is it possible for a line to pass through only one quadrant?
How about four quadrants? explain
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An electrician charges a $50 house call fee and $65 per hour for the work. Which algebraic expression represents this situation to represent the total cost for h hours of work?
Answers
Thus:
$50+$65h
The algebraic expression is Total Cost = 65h + 50.
Explanation:To represent the total cost for h hours of work, we need to consider the house call fee of $50 and the hourly rate of $65. The total cost will depend on the number of hours worked.
We can calculate the total cost using the expression: Total Cost = (Hourly Rate * Number of Hours) + House Call Fee.
Therefore, the algebraic expression that represents this situation is: Total Cost = 65h + 50.
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