The equation that shows the relationship between the age of Danny and Terry is D = T - 6.
Let's represent Danny's age with the variable D and Terry's age with the variable T. We know that Danny is 22 years old, so we can write the equation D = 22.
We also know that Danny is 6 years younger than Terry, so we can write the equation D = T - 6. Now we can substitute the value of D from the first equation into the second equation to find the value of T.
D = T - 6
22 = T - 6
Add 6 to both sides of the equation:
22 + 6 = T
28 = T
So Terry is 28 years old.
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Jacob is practicing the 100 meter dash. The data show his times in seconds.
14, 13, 13.5, 16, 14, 15.5, 14.5
Which box plot shows the distribution of the data?
12
12
+
13
13
14
Time (s)
14
Time (s)
15
15
Median and Quartiles
16
16
+++
12
←
12
13
Level F
13
14
Time (s)
14
Time (s)
15
15
16
16
*what is the answer to this?
The box plot for given minimum, maximum, median, Q1 and Q3 is plotted below.
What is a box and whisker plot?A box and whisker plot—also called a box plot—displays the five-number summary of a set of data. The five-number summary is the minimum, first quartile, median, third quartile, and maximum. In a box plot, we draw a box from the first quartile to the third quartile.
The given data set is,
14, 13, 13.5, 16, 14, 15.5, 14.5.
Hence, We get;
Order of data is,
⇒ 13, 13.5, 14, 14, 14.5, 15.5, 16.
Here, minimum is 13
Maximum is 16
Median is 14
Lower quartile range(Q1) is 13.5
Upper quartile range(Q3) is 15.5
Therefore, The box plot for given minimum, maximum, median, Q1 and Q3 is plotted below.
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Feb 20, 9:49:08 PM Find the axis of symmetry of the parabola defined by the equation x=(1)/(36)y^(2)-(1)/(18)y+(37)/(36).
The axis of symmetry of the parabola defined by the equation x=(1)/(36)y^(2)-(1)/(18)y+(37)/(36) is the vertical line x = (37)/(36). This is because the equation is in the form of y = ax^2 + bx + c, and the axis of symmetry is x = -b/2a. In this case, a = (1)/(36), b = -(1)/(18) and c = (37)/(36).
Therefore, the axis of symmetry is x = -(1)/(36) x -(1)/(18) = (37)/(36). This axis of symmetry divides the parabola into two symmetric halves and it is also the x-coordinate of the vertex of the parabola.
The vertex is the highest or lowest point of the parabola and it is the point of intersection between the axis of symmetry and the parabola. To find the y-coordinate of the vertex, we can substitute the x-coordinate of the vertex into the equation, which gives us y = (1)/(72) + (37)/(36) = (109)/(72). Therefore, the coordinate of the vertex is (37/36, 109/72).
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Question 4. 1. Find the number of non-negative integer solutions ofx1+x2+x3+x4=10. 2. (True/False) : The answer to the above question is same as the number of non-negative integer solutions ofx1+x2+x3≤10. Justify. Is it the same as the number of non-negative integer solutions ofx1+x2+x3+x4+x5=10? Justify. 3. Find the number of positive integer solutions ofx1+x2+x3+x4=10
1. 286
2. False
3. 84
1. The number of non-negative integer solutions of x1+x2+x3+x4=10 is 286.
2. The answer to the above question is not the same as the number of non-negative integer solutions of x1+x2+x3≤10. The reason is that the first equation has 4 variables, while the second equation has only 3 variables.
Therefore, the number of solutions will be different.
3. The number of positive integer solutions of x1+x2+x3+x4=10 is 84.
1. To find the number of non-negative integer solutions of x1+x2+x3+x4=10, we can use the formula: C(n+k-1, k-1) = C(10+4-1, 4-1) = C(13, 3) = 286
Therefore, there are 286 non-negative integer solutions of x1+x2+x3+x4=10.
2. The answer to the above question is not the same as the number of non-negative integer solutions of x1+x2+x3≤10 because the first equation has 4 variables, while the second equation has only 3 variables.
Therefore, the number of solutions will be different.
3. To find the number of positive integer solutions of x1+x2+x3+x4=10, we can use the formula: C(n-1, k-1) = C(10-1, 4-1) = C(9, 3) = 84
Therefore, there are 84 positive integer solutions of x1+x2+x3+x4=10.
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A rectangular prism has a volume of 109.86 cubic centimeters. The width of the prism is 13.2 cm. The height of the prism is 4.1 cm. What is the length of the rectangular prism? Show work.
A. 0.5 cm
B. 2.1 cm
C. 2.03 cm
D. 6.4 cm
The length of the rectangular box will be 2.03 cm. Then the correct option is C.
What is the volume of the rectangular prism?Let the prism with a length of L, a width of W, and a height of H. Then the volume of the prism is given as,
V = L x W x H
The volume of a rectangular box is 109.86 cubic centimeters. The prism's breadth is 13.2 cm. The prism stands 4.1 cm tall.
The length of the rectangular prism is given as,
109.86 = L x 13.2 x 4.1
109.86 = 54.12L
L = 109.86 / 54.12
L = 2.03 cm
The length of the rectangular box will be 2.03 cm. Then the correct option is C.
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please someone respond quickly this is due tonight
Answer:
x=55°, y=35°
Step-by-step explanation:
The angle between x and 35° is a right angle.
A right angle equals 90°.
Therefore. 90=x+35
x=55°
Now that x is known, we can find y.
The angle made between the x and y axis is 90°.
Therefore, 90=x+y
90=55+y
y=35°
Answer:
X=55° and Y=35° ................................
A population of bacteria is decaying.
The number of bacteria after h hours is given by the expression 350(1 - 0.3)
Which statement is true?
A)Each hour, the population decreases by 3%
B)Each hour, the population increase by 3 bacteria.
c) The hourly decay rate for the population is 30%
D)The initial population of bacteria is 245.
Answer:
D
Step-by-step explanation:
Ross has a rectangular garden in his backyard. He measures one side of the garden as 30 feet and diagonal as 33 feet. What is the length of the other side of his garden? Round to the nearest tenth of a foot.
Therefore, the length of the other side of Ross's garden is approximately 13.7 feet.
What is rectangle?A rectangle is a four-sided flat shape with four right angles (90-degree angles) between opposite sides. It is a type of quadrilateral, which means it has four sides. In a rectangle, the opposite sides are equal in length and parallel to each other, which makes it different from a square, where all sides are equal in length. The area of a rectangle can be found by multiplying the length by the width, and the perimeter can be found by adding up the lengths of all four sides.
Here,
Let the length of the other side of the garden be x feet.
Using the Pythagorean theorem, we have:
x² + 30² = 33²
Simplifying the equation:
x² = 33² - 30² = 1089 - 900
= 189
Taking the square root of both sides:
x = √189 ≈ 13.7 feet
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f(x)=2x^3+3x^2+50x+75 find all zeros of polynomial functions
All zeros of polynomial functions -3/2 , 5i and -5i
What is zeros of polynomial ?Zeros of polynomial are the points where the polynomial equals zero on the whole. In simple words, we can say that zeros of polynomial are values of the variable such that the polynomial equals 0 at that point. Zeros of a polynomial are also referred to as the roots of the equation and are often designated as α, β, γ respectively. Some of the methods used to find the zeros of polynomial are grouping, factorization, and using algebraic expressions.
2x^3+3x^2+50x+75 equal to 0
2x^3+3x^2+50x+75 = 0
⇒ x^2 (2x + 3) + 25 (2x + 3) = 0
⇒ (2x + 3) (x^2 + 25 ) = =
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0
2x + 3 = 0
x^2 + 25 = 0
so, x = -3/2 and x = 5i and -5i
The final solution is all the values that make 2x^3+3x^2+50x+75 = 0
x = -3/2 , 5i and -5i
Hence, all zeros of polynomial functions -3/2 , 5i and -5i
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You need to spend exactly $1 and buy 40 stamps. There are 3
kinds of stamps that cost 1, 4, and 12 cents respectively. How many
stamps do you need to buy of each kind (if you have at least one of
each
You need to buy 1 stamp that costs 12 cents, 3 stamps that cost 4 cents each, and 8 stamps that cost 1 cent each, to have a total of 40 stamps and spend exactly $1.
The reasoning is as follows:
If you were to buy only 1 cent stamps, you would need to purchase 100 of them to spend $1, which would exceed the required 40 stamps. On the other hand, if you were to buy only 12 cent stamps, you would only be able to purchase 8 stamps, which would fall short of the required 40 stamps. Therefore, a combination of all three types of stamps is necessary.
Starting with the most expensive stamp, one 12 cent stamp leaves you with 88 cents. Then, 3 stamps of 4 cents each will cost you 12 cents, leaving you with 76 cents. Finally, 8 stamps of 1 cent each will cost you 8 cents, which adds up to $1 and gives you a total of 40 stamps.
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Each expression that has a sum of 4 4/12?
3 1/5+1 3/7
1 11/12+ 2 1/4
1 3/6 + 1 1/2 +1 3/4
1 1/3+ 1 1/2 +1 2/4
3/6+1 2/4+2 1/3
3 1/5 + 1 3/7
To add these two fractions, we need to find a common denominator. The least common multiple of 5 and 7 is 35.
3 1/5 = 16/5
1 3/7 = 10/7
16/5 + 10/7 = (16/5) * (7/7) + (10/7) * (5/5) = 112/35 + 50/35 = 162/35
We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor (GCF), which is 1.
162/35 = 4 22/35
This expression does not equal 4 4/12.
1 11/12 + 2 1/4
Again, we need to find a common denominator to add these two fractions. The least common multiple of 4 and 12 is 12.
1 11/12 = 23/12
2 1/4 = 9/4
23/12 + 9/4 = (23/12) * (1/1) + (9/4) * (3/3) = 23/12 + 27/12 = 50/12
We can simplify this fraction by dividing both the numerator and denominator by their GCF, which is 2.
50/12 = 4 2/12
This expression does equal 4 4/12.
Maria has $30,000 invested in two accounts. Account A earns 4.5% annual interest and account B earns 3.8%. The account earns $1,294 annually. How much does she have invested in each account?
Maria has $22,000 invested in account A and $8,000 invested in account B.
Let's begin by setting up a system of equations to solve for the amount Maria has invested in each account. Let x be the amount invested in account A and y be the amount invested in account B. Since Maria has a total of $30,000 invested in both accounts, we can write the first equation as:
x + y = 30,000
Next, we can write an equation to represent the total annual interest earned from both accounts:
0.045x + 0.038y = 1,294
Now, we can use the substitution method to solve for x and y. We'll rearrange the first equation to solve for x:
x = 30,000 - y
Then, we'll substitute this value of x into the second equation:
0.045(30,000 - y) + 0.038y = 1,294
Simplifying the equation gives:
1,350 - 0.045y + 0.038y = 1,294
Combining like terms and isolating y gives:
0.007y = 56
y = 8,000
Now, we can substitute this value of y back into the first equation to solve for x:
x = 30,000 - 8,000
x = 22,000
So, Maria has $22,000 invested in account A and $8,000 invested in account B.
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browns books are 2 for $11 and noble books are 5 for $31 who’s is cheaper
Therefore , the solution of the given problem of unitary method comes out to be Browns books are less expensive than Noble books at $5.50 each.
What is unitary method ?Take the lengths of this minute subsection and split the sum by two to complete a task using a unitary variable technique. The unit technique, in a nutshell, removes a wanted item both from the specific sets and color subsets. For instance, 40 pencils will cost Rs/kg ($1.01). It's possible that one country will have total influence over the approach taken to accomplish this. Almost every living creature has a distinctive quality. There are unanswered questions and changes (mathematics, algebra).
Here,
We must ascertain the price per book for each retailer in order to evaluate the costs of Browns and Noble books.
Two of Brown's books cost $11, so each volume costs $5.50 (11/2 = 5.50).
Five novels from Barnes & Noble cost $31, making each book $6.20 (31/5 = 6.20).
Browns books are less expensive than Noble books, which cost $6.20 per volume, at $5.50 each.
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Find two numbers a and b such that the following system of linear equations is inconsistent. ax-3y=2 -4x+5y=b
The values of a and b that will result in an inconsistent system are a = 12/5 and b = 20x - 5y.
To find two numbers a and b such that the system of linear equations is inconsistent, we need to make sure that the two equations have the same slope but different y-intercepts. This will result in the two equations being parallel to each other and never intersecting, making the system inconsistent.
The first equation is: ax - 3y = 2. We can rearrange this equation to solve for y and find the slope:
3y = ax - 2
y = (a/3)x - (2/3)
The slope of this equation is a/3.
The second equation is: -4x + 5y = b. We can rearrange this equation to solve for y and find the slope:
5y = 4x + b
y = (4/5)x + (b/5)
The slope of this equation is 4/5.
For the system to be inconsistent, the slopes need to be equal. So we can set a/3 = 4/5 and solve for a:
a/3 = 4/5
a = 12/5
Now we need to find a value for b that will result in a different y-intercept. We can do this by plugging in the value of a into one of the equations and choosing a different value for the y-intercept:
y = (12/5)(1/3)x - (2/3)
y = 4x - (2/3)
If we choose a different value for the y-intercept, such as -1, we can solve for b:
y = 4x - 1
5y = 20x - 5
b = 20x - 5y
So the values of a and b that will result in an inconsistent system are a = 12/5 and b = 20x - 5y.
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What is the slope of a line, in the standard (x,y) coordinate plane, that is parallel to x+5y=9 ?
Answer:-15
Step-by-step explanation:
−3(0.75x−2y)+6(0.5x−2y) ?
How many rebounds should your team expect to have in 15 missed shots?
It is unlikely that the projected rebounds in 15 shots will actually happen.
Explain about the term probability?A probability is a scaled version of the likelihood or chance that a specific event will take place. Both percentages with values ranging from zero to one and percentages extending from 0% to 100% can be used to describe probabilities.Total shots fired equals 10.
To find that there were 15 shots total.
Consider S to be the likelihood that your team will rebound and miss a shot.
P(S) = Number of rebounds / Total Number of outcomes
P(S) = 7/15
P(S) = 0.46
Consequently, it is unlikely that the projected rebounds in 15 shots will actually happen.
Thus, it is unlikely that the projected rebounds in 15 shots will actually happen.
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The complete question is-
During a basketball game, you record the number of rebounds from missed shots for each team. How many rebounds should your team expect to have in 15 missed shots?
Picture for question is attached.
Solve for x. Assume that lines which appear to be diameters are actual diameters.
Assuming the lines, which appear to be diameters, are actual diameters. The value of x is 10.63.
What is the diameter?Any straight line that connects two points on a circle's circumference and goes through the circle's center. The length of such a line in geometry. The greatest separation possible between any two points in a metric space (geometry).
Exterior angles are 128, 32, and 45
Interior angles are 11x+4 and 85
Exterior angles = interior angles
128 + 32 + 45 = 11x+4 + 85
205 = 11x + 88
11x = 205 - 88 = 117
x = 117/11
x = 10.63
Therefore, the value of x is 10.63.
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name the following polynomial by its degree and number of the terms. then prove its degree by using successive differences g(x)=x^(3)+3x^(2)-x-3
The polynomial g(x)=x^(3)+3x^(2)-x-3 is a cubic polynomial with 4 terms.
To prove its degree using successive differences, we first need to find the successive differences of the polynomial's y-values for consecutive x-values. We can do this by substituting x-values into the polynomial and finding the differences between the resulting y-values.
For x=0, g(x)=0^(3)+3(0^(2))-0-3=-3
For x=1, g(x)=1^(3)+3(1^(2))-1-3=0
For x=2, g(x)=2^(3)+3(2^(2))-2-3=11
For x=3, g(x)=3^(3)+3(3^(2))-3-3=30
The first successive difference is 0-(-3)=3
The second successive difference is 11-0=11
The third successive difference is 30-11=19
Since the successive differences are not constant, we need to find the successive differences of the successive differences.
The first successive difference of the successive differences is 11-3=8
The second successive difference of the successive differences is 19-11=8
Since the successive differences of the successive differences are constant, the degree of the polynomial is 3, which is one less than the number of times we had to find successive differences. This confirms that g(x)=x^(3)+3x^(2)-x-3 is a cubic polynomial.
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5(y+25)=−13
y = ??
Please answer this if you can!!
Answer:
y = - 125/18
Step-by-step explanation:
okay so here what i got when i was doing the equation
5 ( y + 25) = - 13y
5 y + 125 = - 13y
then you subtract 125 from both sides
subtract the numbers
and my solution was y = - 125 / 18 trust me!
Hope that helped
Rewrite the set U by listing its elements. Make sure to use the appropriate set notation. U={z|z is an integer and -3<=z<=-1}
The answer of set U by listing its elements is {-3, -2, -1}
To rewrite the set U by listing its elements, we need to identify the integers that fall within the given range of -3<=z<=-1.
The appropriate set notation for listing the elements of a set is {element1, element2, element3, ...}.
So, the integers that fall within the given range are -3, -2, and -1.
Therefore, we can rewrite the set U as:
U = {-3, -2, -1}
This is the answer in the appropriate set notation, listing the elements of the set U.
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Solve the given inequality and graph the solution on a number line
-x/2+3/2<5/2
Graph on number line
Fοr the inequality the sοlutiοn set is x > -2.
What is an inequality?In Algebra, an inequality is a mathematical statement that uses the inequality symbοl tο illustrate the relatiοnship between twο expressiοns. An inequality symbοl has nοn-equal expressiοns οn bοth sides. It indicates that the phrase οn the left shοuld be bigger οr smaller than the expressiοn οn the right, οr vice versa.
Tο sοlve the inequality, we can fοllοw these steps -
Subtract 3/2 frοm bοth sides -
-x/2 < 5/2 - 3/2
-x/2 < 2/2
-x/2 < 1
Multiply bοth sides by -2 (and flip the inequality since we're multiplying by a negative number) -
x > -2
The graph is drawn fοr the sοlutiοn.
The circle οn the left endpοint is nοt filled because the inequality is strict (i.e., nοt less than οr equal tο).
The arrοw οn the right endpοint is filled because the sοlutiοn set includes all values greater than -2.
Therefοre, the sοlutiοn set is all real numbers greater than -2, which can be represented οn a number line as a ray starting at -2 and mοving tοwards pοsitive infinity.
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The total surface area of a spherical segment is (7) times greater than the surface area of the sphere inscribed in it. Determine the altitude of the segment. if the radius of its spherical surface is equal to R.
To find the altitude of the spherical segment, we need to use the formula for the total surface area of a spherical segment, which is given by:
A = 2πR(r + h), where R is the radius of the spherical surface, r is the radius of the base, and h is the height of the segment.
We are given that the total surface area of the spherical segment is 7 times greater than the surface area of the sphere inscribed in it, which means that:
7(4πR^2) = 2πR(r + h)
Simplifying this equation gives us:
14R = r + h
We are also given that the radius of the spherical surface is equal to R, which means that r = R. Substituting this into the equation gives us:
14R = R + h
Solving for h, we get:
h = 14R - R
h = 13R
Therefore, the altitude of the spherical segment is 13R.
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A box contains a penny, a nickel, and a dime.
Find the probability of choosing a dime first and without replacing the dime, choosing a penny.
A 1/6
B 1/7
C 1/8
D 1/9
Answer:When we choose a dime from the box, there are two coins left, one penny and one nickel. Since we do not replace the dime, the probability of choosing a penny next is 1/2.
Therefore, the probability of choosing a dime first and a penny second is:
P(dime first and penny second) = P(dime first) * P(penny second | dime first)
= (1/3) * (1/2)
= 1/6
So, the answer is A. 1/6.
Step-by-step explanation:
Three friends are shopping at the garage sale shown.
Items for a garage sale are shown. The price of pants is eight dollars. The price of shirts is six dollars. The price of shorts is four dollars. The price of belts is three dollars.
Camille can buy up to 5 shirts. How much money could she have?
Enter the correct answers in the boxes in dollars and cents.
She has at least $
and at most $
The minimum amount of money Camille could have is 0 dollars if she doesn't buy anything, and the maximum amount of money she could have is 166 dollars if she buys the maximum quantity of each item.
How to obtain the maximum value of a function?To find the maximum of a continuous and twice differentiable function f(x), we can firstly differentiate it with respect to x and equating it to 0 will give us critical points.
Putting those values of x in the second rate of function, if results in negative output, then at that point, there is maxima. If the output is positive then its minima and if its 0, then we will have to find the third derivative (if it exists) and so on.
We are given that;
The prices of different items at a garage sale: pants, shirts, shorts, and belts.
Camille can buy up to 5 shirts.
Now,
If Camille buys 5 shirts, she would spend 5 * 6 = 30 dollars on shirts. Since she can buy up to 5 shirts, she may spend less than 30 dollars if she buys fewer shirts.
She can buy up to 10 pants, which would cost her 10 * 8 = 80 dollars.
She can buy up to 5 shorts, which would cost her 5 * 4 = 20 dollars.
She can buy up to 12 belts, which would cost her 12 * 3 = 36 dollars.
30 + 80 + 20 + 36 = 166 dollars
Therefore, by the maxima and minima the answer will be 166 dollars if she buys the maximum quantity of each item.
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If 1760 litres of fuel is sold in 5 days, in how many days will 3872 litres be sold
It will take 11 days to sell 3872 liters of fuel at the same rate as 1760 liters sold in 5 days.
We can use the following proportion to solve the problem:
1760 liters / 5 days = 3872 liters / x days
Where x is the number of days it will take to sell 3872 liters of fuel.
We can use the unitary method to solve this problem.
Let's start by finding out how much fuel is sold in one day. To do this, we need to divide the total amount of fuel sold (1760 liters) by the number of days it was sold for (5 days):
1760 liters ÷ 5 days = 352 liters per day
Now we can use this rate to find out how many days it will take to sell 3872 liters of fuel:
3872 liters ÷ 352 liters per day = 11 days (rounded to the nearest whole number)
Therefore, it will take 11 days to sell 3872 liters of fuel at the same rate as 1760 liters sold in 5 days.
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3. Solve each equation.
a. 2(x-3) = 14
X=10
b. -5(x - 1) = 40
DATE
c. 12(x + 10) = 24
d. (x + 6) = 11
The value of x for the given equations are a. 10, b. -7, c = -8, d. 5.
What are equations and its solution?Finding an equation's solutions, which are values (numbers, functions, sets, etc.) that satisfy the equation's condition and often consist of two expressions connected by an equals sign, is known as solving an equation in mathematics. One or more variables are identified as unknowns when looking for a solution. An assignment of values to the unknown variables that establishes the equality in the equation is referred to as a solution. Particularly but not exclusively for polynomial equations, the solution of an equation is frequently referred to as the equation's root.
Given that,
2(x-3) = 14
2x - 6 = 14
2x = 20
x = 10
b. -5(x - 1) = 40
-5x + 5 = 40
-5x = 35
x = -7
c. 12(x + 10) = 24
12x + 120 = 24
12x = 24 - 120
x = -8
d. (x + 6) = 11
x = 11 - 6
x = 5
Hence, the value of x for the given equations are a. 10, b. -7, c = -8, d. 5.
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>If √x-3a+√x-3b=√x-3c then prove that x=(a+b+c) ±2√a² + b² + c²-ab-bc-ca
Answer: See the Picture.
Step-by-step explanation:
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Vincent and some friends shared the cost of a season ticket package for the local football team. The package cost $745 and each person contributed $186.25. Write a multiplication equation that can be
The multiplication equation that can be used to determine the number of people who contributed to the cost of the season ticket package is: $745 = $186.25 x 4.
To write a multiplication equation that can be used to determine the number of people who contributed to the cost of the season ticket package, we can use the following equation:
Total Cost = Cost per Person x Number of People
In this case, the total cost is $745 and the cost per person is $186.25. We can plug these values into the equation to get:
$745 = $186.25 x Number of People
To solve for the number of people, we can divide both sides of the equation by $186.25:
Number of People = $745 / $186.25
Number of People = 4
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what is the vaule of z??
Answer:
Z= 6
Step-by-step explanation:
Z + 9 = 15
- do the inverse (opposite) operation
15 -9 = 6
9 -9 * cancel out *
Z = 6
CHECK:
6 +9 = 15
TRUE
Find the missing length indicated. Leave your answer in simplest radical form.
Answer:
x = 16
AB = 25
BC = 15
AC = 20
Step-by-step explanation:
I have marked the vertices to explain
There are three right triangles that can be seen in the picture
Δ ABC with legs BC and AC and hypotenuse AB
ΔACD with legs AD and CD and hypotenuse AC
ΔBCD with legs CD and BD and hypotenuse BC
We will using the Pythagorean formula in all three triangles:
hypotenuse² = sum of squares of legs
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(1) Let's first find missing side BC
In right triangle ΔBCD ,
BC² = CD² + BD²
We have CD = 12, BD = 9
BC² = 12² + 9²
= 144 + 81
= 225
BC = √225 = 15
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(2) Now let's focus on ΔABC
AB is the hypotenuse
AC and BC are the legs
AB² = AC² + BC²
or
AC² = AB² - BC²
Since AB = AD + BD = x + 9 and BC = 15
AC² = (x + 9)² - 15²
(x + 9)² = x² + 2 · 9 · x + 9² = x² + 18x + 81
15² = 225
AC² = x² + 18x + 81 -225
AC² = x² + 18x - 144 (1)
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(3) Now let's turn our attention to ΔACD with legs AD and CD and hypotenuse AC
AC² = AD² + CD²
With AD = x and CD = 12,
AC² = x² + 12²
AC² = x² + 144 (2)
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Equations (1) and (2) have the same left side, AC²
So the expressions on the right side must be equal
Therefore:
x² + 18x - 144 = x² + 144
x² terms cancel out from left and right sides giving
18x - 144 = 144
Add 144 on both sides:
18x - 144 + 144 = 144 + 144
18x = 288
x = 288/18 = 16
Plugging this into equation (2) AC² = x² + 144 gives
AC² = 16² + 144
AC² = 256 + 144
AC² = 400
AC = √400
or
AC = 20
Since AB is the third unknown side and AB = x + 9 we get
AB = 16 + 9 = 25