Answer: 14π
Step-by-step explanation:
The area of a circle can be expressed as πr^2. In this case, the area is given as 49π m², so we can write:
πr^2 = 49π
Dividing both sides by π, we get:
r^2 = 49
Taking the square root of both sides, we get:
r = 7
The circumference of a circle is given by 2πr, so the circumference in this case would be:
C = 2π * 7 = 14π
So the circumference of the circle with an area of 49π m² is 14π meters.
Please answer these for 100 points
The area of a trapezoid with the given parallel sides and height is 4x+8 square units. Therefore, option B is the correct answer.
What is area of the trapezoid?The area of a trapezoid can be calculated if the length of its parallel sides and the distance (height) between them is given. The formula for the area of a trapezoid is expressed as, A = ½ (b₁+b₂)×h.
where (A) is the area of a trapezoid, 'b₁' and 'b₂' are the bases (parallel sides), and 'h' is the height (the perpendicular distance between b₁ and b₂).
1) Given that, b₁=x-3, b₂=x+7 and the height=4
Now, area of a trapezoid = 1/2 (x-3+x+7)×4
= 2×(2x+4)
= 4x+8 square units
Hence, option B is the correct answer.
2) The given polynomials are -3x²-7x+9 and -5x²+6x-4.
Here, sum = -3x²-7x+9+(-5x²+6x-4)
= -3x²-7x+9-5x²+6x-4
= -8x²-x+5
Hence, option 1 is the correct answer.
Therefore, the area of a trapezoid is 4x+8 square units.
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You are on a team of architects. You are charged with building a scale-model replica of one section of a new roller coaster before construction gets underway.
Certain reinforcement cables and struts are required to make the roller coaster sturdier. The goal for this project is for your team to determine where to place these cables or struts. The mathematical models for these reinforcements are known.
Your team must provide both algebraic and graphical evidence for your conclusions regarding the location of the cables.
Directions:
The shape of this particular section of the rollercoaster is a half of a circle. Center the circle at the origin and assume the highest point on this leg of the roller coaster is 30 feet above the ground.
1. Write the equation that models the height of the roller coaster.
The circle equation is x² + y² = r²
Start by writing the equation of the circle. (Recall that the general form of a circle with the center at the origin is x2 + y2 = r2. (10 points)
The roller coaster's leg is 30 feet above the ground, as stated.
⇒ r = 30
⇒x² + y² = 30²
⇒x² + y² = 900
Now solve this equation for y. Remember the roller coaster is above ground, so you are only interested in the positive root. (10 points)
Roller coaster height equation: - x2 + y2 = 900
As we know, x² + y² = 900
⇒y² = 900 - x²
⇒y = √900 - x²
2. Graph the model of the roller coaster using the graphing calculator. Take a screenshot of your graph and paste the image below, or sketch a graph by hand. (5 points)
Model 1: One plan to secure the roller coaster is to use a chain fastened to two beams equidistant from the axis of symmetry of the roller coaster, as shown in the graph below:
You need to determine where to place the beams so that the chains are fastened to the rollercoaster at a height of 25 feet.
3. Write the equation you would need to solve to find the horizontal distance each beam is from the origin. (10 points)
To find the horizontal distance, we use the equation: x = 30^2-25^2
4. Algebraically solve the equation you found in step 3. Round your answer to the nearest hundredth. (10 points)
The horizontal distance is 16.58 feet
How to determine the equation:
The height is given as: 25 feet.
The roller coaster's length is 30 feet.
Use h to represent horizontal distance.
Given that the roller coaster represents the hypotenuse side length, the following equation can be used to calculate h
In step 3, we have: 30^2-25^2
This gives
Take the roots of both sides to get our answer
5. Explain where to place the two beams. (10 points)
The beam should be placed 8 feet from the center.
The struts are y = √(x + 8) and y = √(x − 4).
The struts are 2 feet apart at the location of the beam: √(x + 8) − √(x − 4) = 2
Solving:
√(x + 8) = 2 + √(x − 4)
x + 8 = 4 + 4√(x − 4) + x − 4
8 = 4√(x − 4)
2 = √(x − 4)
x − 4 = 4
x = 8
Model 2: Another plan to secure the roller coaster involves using a cable and strut. Using the center of the half-circle as the origin, the concrete strut can be modeled by the equation and the mathematical model for the cable is. The cable and the strut will intersect.
6. Graph the cable and the strut on the model of the roller coaster using the graphing calculator. Take a screenshot of your graph and paste the image below, or sketch a graph by hand. (5 points)
7. Algebraically find the point where the cable and the strut intersect. Interpret your answer. (10 points)
root 2x+8=x-8
2x+8=x^2-16x+64
-x^2+18x-56=0
x^2-18x+56=0
x(x-4)-14(x-4)=0
(X-4)(x-14)=0
x=4 ,x=14
x=14
Model 3: Another plan to secure the roller coaster involves placing two concrete struts on either side of the center of the leg of the roller coaster to add reinforcement against southerly winds in the region. Again, using the center of the half-circle as the origin, the struts are modeled by the equations and. A vertical reinforcement beam will extend from one strut to the other when the two cables are 2 feet apart.
8. Graph the two struts on the model of the roller coaster. Take a screenshot of your graph and paste the image below, or sketch a graph by hand. (5 points)
Recall that a reinforcement beam will extend from one strut to the other when the two struts are 2 feet apart.
9. Algebraically determine the x -value of where the beam should be placed. (15 points)
10. Explain where to place the beam. (10 points)
Step 1: Equation for Roller Coaster Height.
The general form for the equation of a circle at the origin is:
[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Equation of a Circle (at origin):}}\\\\x^2+y^2=r^2\\\\\bullet \ \text{where 'r' is the radius of the circle}\end{array}\right}[/tex]
It is given that the height of the roller coaster is 30 feet. Thus, r = 30. Plug this into the equation above and solve for 'y.'
[tex]\Longrightarrow x^2+y^2=(30)^2\\\\\\\\\Longrightarrow x^2+y^2=900\\\\\\\\\therefore \boxed{\boxed{y=\sqrt{900-x^2} }}[/tex]
Step 2: Graph of Roller Coaster Model.
Refer to the attached image(s).
Step 3: Equation for Horizontal Distance of Beams.
Going back to the equation we got in step (1), we can solve for the variable ‘x.’
[tex]\Longrightarrow y=\sqrt{900-x^2}\\\\\\\\\therefore \boxed{\boxed{x=\pm\sqrt{900-y^2}}}[/tex]
Step 4: Algebraic Solution for Beam Placement.
Given a height, y = 25 ft. We can plug this into the equation above and solve for 'x.'
[tex]\Longrightarrow x=\pm\sqrt{900-(25)^2}\\\\\\\\\Longrightarrow x=\pm\sqrt{275}\\\\\\\\\Longrightarrow x=\pm5\sqrt{11}\\\\\\\\\therefore \boxed{\boxed{x\approx\pm16.58 \ ft}}[/tex]
Step 5: Placement of Beams.
The beams should be placed 16.58 feet from either side of the origin.
Step 6: Graph of Cable and Strut.
Refer to the attached image(s).
Step 7: Algebraic Intersection of Cable and Strut.
To find the point of intersection, set the two given equations (from model 2) equal to each other and solve for 'x.'
[tex]y=\sqrt{2x+8} \ \text{and} \ y=x-8\\ \\\\\\\Longrightarrow \sqrt{2x+8}=x-8 \\\\\\\\\Longrightarrow 2x+8=(x-8)^2 \\\\\\\\\Longrightarrow 2x+8=x^2-16x+64\\\\\\\\\Longrightarrow x^2-18x+56=0\\\\\\\\\Longrightarrow (x-14)(x-4)=0\\\\\\\\\therefore x=\{14,-4\}[/tex]
In order to determine which value of 'x' is correct we must verify the solution.
When x = 14:
[tex]\Longrightarrow 2(14)+8=(14-8)^2\\\\\\\\\Longrightarrow 36=36 \checkmark[/tex]
When x = -4:
[tex]\Longrightarrow 2(-4)+8=(-4-8)^2\\\\\\\\\Longrightarrow 0\neq 144[/tex]
Thus, the correct x-value is 14. Plug this into either given equations and solve for 'y.'
[tex]\Longrightarrow y =x-8\\\\\\\\\Longrightarrow y =14-8\\\\\\\\\therefore y=6[/tex]
Thus, the point of intersection is (14, 6).
Step 8: Graph of Two Struts.
Refer to the attached image(s).
Step 9: Algebraic Solution for Beam Placement.
To find the x-value of the beam placement, you need to find the point where the two strut equations are 2 feet apart. Set up an equation using the given functions and solve for 'x.'
[tex]\Longrightarrow y=\sqrt{x+8}-\sqrt{x-4}; \ y=2\\\\\\\\\Longrightarrow 2=\sqrt{x+8}-\sqrt{x-4}\\\\\\\\\Longrightarrow 2+\sqrt{x-4}=\sqrt{x+8}\\\\\\\\\Longrightarrow 4\sqrt{x-4}+x=x+8\\\\\\\\\Longrightarrow 4\sqrt{x-4}=8\\\\\\\\\Longrightarrow \sqrt{x-4}=2\\\\\\\\\Longrightarrow x-4=4\\\\\\\\\therefore \boxed{\boxed{x=8}}[/tex]
Step 10: Placement of Beam.
The beams will need to be placed 8 feet from either side of the origin.
The ratio of staff to guests at the gala was 3 to 5. There were a total of 576 people in the ballroom. How many guests were at the gala?
360 guests were at the gala
How to determine how many guests were at the gala?Ratio is used to compare two or more quantities. It is used to indicate how big or small a quantity is when compared to another.
Since the ratio of staff to guests at the gala was 3 to 5.
Let S and G represent the number of staff and guests respectively. From the given information, we can write:
S/G = 3/5
S + G = 576
Thus, S = 576 - G
Put S = 576 - G in S/G = 3/5 and solve for G. That is:
(576 - G)/G = 3/5
3G = 5(576 - G)
3G = 2880 - 5G
3G + 5G = 2880
8G = 2880
G = 2880/8
G = 360
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PLEASE ONLY ANSWER IF YOU KNOW
The value of AB in the triangle is 2.5 cm and the value of x in the triangle is 10 units
How to determine the value of ABFrom the question, we have the following parameters that can be used in our computation:
The triangle
On the triangle, we have the following equivalent ratio
AB : 4 = 10 : 16
Express as fraction
So, we have
AB/4 = 10/16
This gives
AB = 4 * 10/16
Evaluate
AB = 2.5
How to determine the value of xHere, we understand that the line bisects the angle BAC
This means that
x = 20 - x
So, we have
2x = 20
Divide
x = 10
Hence, the value is 10
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Find the 11th term of the geometric sequence 7, 35, 175
11th term of the given geometric sequence is [tex]a_{n} = 7(5^{n-1})[/tex]
What is Geometric sequence?In a Geometric Sequence each term is found by multiplying the previous term by a constant.
Given geometric sequence = 7,35,175
Nth term of the geometric sequence [tex]a_{n}=a_{1} .r^{n-1}[/tex]
[tex]a_{1}=7[/tex]
[tex]r= \frac{35}{7} =5[/tex]
Nth term of the geometric sequence
[tex]a_{n} = 7(5^{n-1})[/tex]
Hence, the Nth term of the given geometric sequence is [tex]a_{n} = 7(5^{n-1})[/tex]
If 14 cars and 10 vans are in a parking lot what is the radio of cars to vans
Answer: 7:5
Step-by-step explanation:
Drag each tile to the correct box. Not all tiles will be used.
Order the steps for bisecting line segment AB using a reflective device.
A
Place the reflective device anywhere on
the segment.
Move the reflective device around on the
paper until the reflection of point A
coincides with point B.
Place the reflective device on Point A.
Place the reflective device on Point B.
Draw the line of reflection using the edge
of the reflective device as a guide.
The correct steps for bisecting line segment AB using a reflective device is given below:
Place the reflective device on Point A.Move the reflective device around on the paper until the reflection of point A coincides with point B.Draw the line of reflection using the edge of the reflective device as a guide.What is Bisecting a line?Cutting a line exactly in half is known as bisecting it. Due to the fact that the line you are drawing will be at a right angle to the initial line, it is also sometimes referred to as creating a perpendicular bisector.
A compass, pencil, and ruler are required.
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What is the slope of a line perpendicular to a horizontal line?
The slope of a horizontal line perpendicular to a horizontal line is infinity or undefined.
The slope of a line is the that of the inclination that is makes with a horizontal line.
So, the slope of horizontal line is 1 because the value of tam(0 degrees) is also 1.
A line that is perpendicular to the horizontal line will have a the value of angle to be 90 degrees and we know that,
tan (90 degrees) = undefined/infinity
So, here we are concluding that the slope of any line that is being lying perpendicularly on any horizontally lying line is infinity or undefined.
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Sarah needs to earn a C in her Algebra class. Her current test scores are 69, 75, 65, and 83. Her final exam is worth 4 test scores. In
order to earn a C Sarah's average must lie between 70 and 79 inclusive. What range of scores can Sarah receive on the final exam to
earn a C in the course?
s final exam scores[
(Type an integer or a simplified fraction.)
Answer:
Sarah must get between a 67 and 85.
Step-by-step explanation:
69 + 75 + 65 + 83 = 292
70 ≥ 1/8(292 + 4x) ≥ 79
70 ≥ 0.5x + 36.5 ≥ 79
67 ≥ x ≥ 85
a wheel for a wheel of winning game makes one revolution in 4 seconds. What is their unit rate in terms of minutes
It will take 15 revolutions per minute.
What is the unit rate?
An item's unit rate is its price for one of them. This is expressed as a ratio with a one as the denominator. For instance, if you covered 70 yards in 10 seconds, you covered 7 yards on average every second. Seven yards in one second and 70 yards in ten seconds are both ratios, but only one of them is a unit rate.
Here, we have
Given: a wheel for a wheel of winning game makes one revolution in 4 seconds.
Take a time for 1 revolution = 4 seconds
Assume that the unit rate in terms of revolutions per minute is x.
1 rev/4 second = x rev/60 second
4x = 60
x = 15
Hence, it will take 15 revolutions per minute.
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This circle iS centered at the origin, and the length of its radius iS 5. What is
the equation of the circle?
Answer:
25
Step-by-step explanation:
The equation of a circle centered at the origin with radius r is given by:
(x^2) + (y^2) = r^2
In this case, the circle is centered at the origin and has a radius of 5, so the equation is:
x^2 + y^2 = 25
Therefore, the equation of the circle is: x^2 + y^2 = 25
A B What is the converse of the following statement? "If I studied for it, then I did well on the exam." "If I did well on the exam, then I studied for it." "If I did not do well on the exam, then I did not study for it."
Answer:
A
I believe because p→q
q→p
My age 2 years ago plus my age 2 years from now equals 24. How old am I
Answer:
= 6 years old.
Step-by-step explanation:
Let your age 2 year ago be = x
= 2x
again
Let your age 2 years from now be x
= 2x
therefore
= 2x +2x = 24
= 4x = 24
= 4x/4 = 24/4
= x = 6
therefore you're 6 years old.
Find the coordinates of the circumcenter of ABC with vertices A(-9, -4),
B(-9, 8), and C(-1,0).
The coordinates of the circumcenter are?
Answer:
Step-by-step explanation: 32 because 9 x4 lol
PLEASE HELP AS FAST AS POSSIBLE
Answer:
l is parallel to m and u is parallel to t
Step-by-step explanation:
Parallel line means that never meet each other
(3b + 3)(-b² - 2b + 2)
Thomas won $16,000 on a game show, and he put all of the money in a checking account (which does not give interest). Each month, he takes $400 out of the account for "spending" money.
The equation that describes this situation is M=16000−400t
, where t
is the time in months since opening the account, and M
is the amount of money left in the account.
In how many months will Thomas empty the account (which means that there will be no money left)?
It will take Thomas 40 months to empty the account if he continues to take out $400 each month of $16,000.
To find out how many months it will take for Thomas to empty the account, we need to set M to 0 and solve for t. This will give us the number of months it will take for Thomas to spend all of the $16,000 he won on the game show. The equation we need to solve is:
0 = 16000 - 400t
First, we'll isolate the variable t on one side of the equation by adding 400t to both sides:
400t = 1600
Next, we'll divide both sides of the equation by 400 to get the value of t:
t = 16000/400
Simplifying the right side of the equation gives us:
t = 40
So, it will take Thomas 40 months to empty the account if he continues to take out $400 each month for spending money.
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Can someone please help me with this?
Show work please
The closet square inch that is needed to cover the cuboid is 208 inches ( option A)
What is surface area of a cuboid?A cuboid is defined as a three-dimensional shape, that has six rectangular faces, eight vertices and twelve edges.
The surface area of cuboid is given as;
SA = 2(lb+lh+bh)
lb = 33/4 × 4 = 33 in
lh = 23/4 × 4 = 23 in
bh = 23/4 × 33/4 = 759/16
SA = 2( 23+33+759/16)
SA = 46+66+47.44
SA = 207 in
therefore the closet square inches is 208 in²
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Determine the interval(s) on which the given function is increasing.
Polynomial going up from the left and passing through the point negative 1 comma 0 and going to a relative maximum at the point 0 comma 5 and then going down to a relative minimum at the point 1 comma 4 and then going up to the right
(–∞, –1) ∪ (1,∞)
(–1, ∞)
(–∞, 0) ∪ (1, ∞)
(0, 1)
The intervals on which the function is increasing is (0, 1). Option D
What is a polynomial?A polynomial is a type of mathematical expression that has variables that are raised to whole number powers.
Since the factors of 5 are 1 and 5, we'll try to factor the polynomial by assuming that (x-1) and (x-5) are factors.
The product of (x-1) and (x-5) is (x² - 6x + 5), which is the polynomial we're given.
⇒ (x² - 6x + 5) is the factored form of the polynomial.
A polynomial is increasing on the intervals where it's greater than zero. So we just need to look at the sign of (x² - 6x + 5) for values between 1 and 5.
For values between 1 and 5, the function is always positive, so it's increasing on the interval (1, 5). Therefore, the increasing interval is (1, 5).
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Rafael runs 4 miles in 30 minutes. At the same rate, how many minutes would he take to run 10 miles?
Rafael will take 75 minutes to run 10 miles at the same rate.
Solution:
We know that Rafael can run 4 miles in 30 minutes.
So, we can find his pace, as follows:
30 minutes/ 4 miles = 7.5 minutes per mile.
So, for 10 miles he will take 7.5* 10 = 75 minutes.
Hence, it will take Rafael 75 minutes to run 10 miles.
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What is the percent composition of water in the compound magnesium sulfate heptahydrate, MgDO4•7H2O?
A. 7.3%
B. 24.8%
C. 48.8%
D. 51.2%
Answer:51. 2%
Step-by-step explanation:
24+32+(16×4)+7(2+16)=24+32+64+126=246
26g of Epsom salt contains 126g of water of crystallisation.Hence, 100g of Epsom salt contains 100×126/246The % of H2O in MgSO4.7H2O=
What is the answers to 3/4 x 3/4
Answer: 9/16
Step-by-step explanation: Also known as (3/4)^2
1. 3x3/4x4
2. 9/4x4
3. 9/16
You can't simplify anymore because it is in the simplest form already.
help fast need done pls.
my brain is playing not smart on me right now
Find the value of each variable provide proofs
The value of x is 18.
The value of y is 15.6.
What is the Pythagorean theorem?Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
We have,
From the figure,
Sin 30 = 9/x
1/2 = 9/x
x = 9 x 2
x = 18
Now,
Using the Pythagorean theorem.
x² = 9² + y²
18² = 81 + y²
y² = 325 - 81
y = √244
y = 15.6
Thus,
x = 18
y = 15.6
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Write an algebraic expression to represent the following situation: "Greg has nickels and pennies in his pocket. The number of pennies is 7 less than twice the number of nickels. Let N represent the number the nickels. Write an expression for the number of pennies."
Algebraic expression to represent the given situation would be, P = 2N - 7
What is algebraic expression?
When we utilize numbers and words to solve a mathematical problem, we are using algebraic expressions.
Let P represent the number of pennies in Greg's pocket.
We know that the number of pennies is 7 less than twice the number of nickels, which can be written as:
P = 2N - 7
This expression represents the number of pennies in terms of the number of nickels.
Therefore, Algebraic expression to represent the given situation would be, P = 2N - 7
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What is 4.25 in expanded form?
Answer:
4 + 0.2 + 0.05
Step-by-step explanation:
Find any points of discontinuity for the rational function. y=(x+3)(x-5)(x+7)/(x+1)(x+4)
Points of discontinuity for the rational function y=(x+3)(x-5)(x+7)/(x+1)(x+4) are x = -1 and x = -4.
A rational function is a function that can be expressed as the ratio of two polynomial functions. In the case of the given rational function y=(x+3)(x-5)(x+7)/(x+1)(x+4), the numerator is a polynomial of degree 3, and the denominator is a polynomial of degree 2.
To find any points of discontinuity for a rational function, we need to determine where the denominator is equal to zero. Division by zero is undefined, so any value of x that makes the denominator equal to zero will be a point of discontinuity for the function.
In this case, we set the denominator (x+1)(x+4) equal to zero and solve for x:
(x+1)(x+4) = 0
This equation is true if either (x+1) = 0 or (x+4) = 0. So we have two solutions:
x+1 = 0 or x+4 = 0
x = -1 or x = -4
These values of x make the denominator of the function equal to zero, which means that the function is undefined at these points. These points are known as vertical asymptotes of the function, since the function approaches infinity as x approaches these points from either side.
It's important to note that the numerator of the function does not affect the points of discontinuity, since it is always defined for any value of x. Therefore, the points of discontinuity only depend on the denominator of the function.
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The lengths of two sides of a triangle are 4 inches and 7 inches.
What is a possible length, in inches, of the third side?
A possible length, in inches, of the third side is equal to 4 inches.
What is the triangle inequality theorem?In Euclidean geometry, the Triangle Inequality Theorem states that the sum of any two sides of a triangle must be greater than or equal (≥) to the third side of the triangle.
Mathematically, the Triangle Inequality Theorem is represented by this mathematical expression:
b - c < n < b + c
Where:
n, b, and c represent the side lengths of this triangle.
By applying the Triangle Inequality Theorem, the possible side lengths of the triangle's third side is given by:
b - c < n < b + c
7 - 4 < n < 7 + 4
3 < n < 11.
Therefore, n could be 4, 5, 6, 7, 8, 9, and 10 inches.
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write a explicit formula in slope intercept form.
an=-3+5(n-1)
An explicit formula in slope intercept form for the given expression is aₙ = 5n - 8.
What is the slope-intercept form?Mathematically, the slope-intercept form of the equation of a straight line is given by this mathematical expression;
y = mx + c
Where:
m represents the slope or rate of change.x and y are the points.c represents the y-intercept or initial value.Based on the information provided above, we can logically deduce that an equation in slope-intercept form that models the situation is given by:
aₙ = -3 + 5(n - 1)
aₙ = -3 + 5n - 5
aₙ = 5n - 8
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Help with this PLEASE!!
Answer:
x = 24°
Step-by-step explanation:
See the attached worksheet. Let a and b stand for the two unknown angales as shown on the worksheet. Since the sum of angles across a straight line is 180°, we can express the two angles as:
a = 180°-x, and
b = 180° - 111° or b = 69°
The sum of the interior angles of a quadrilateral is 360°. Set the four angles equal to 360° and solve for x.
x = 24°