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Ad libitum [116K]
3 years ago
15

In remodeling a house an architect finds that by adding the same amount to each dimension of a 15ft by 19ft rectangular room, th

e area would be increased by 98 ft^2. How
much must be added to each dimension?
Let x be the amount that is added to each dimension. After writing an equation in standard form with a > 0, a= ? b= ? and c= ?
(Simplify your answers.)
Mathematics
1 answer:
earnstyle [38]3 years ago
8 0

Answer:

Step-by-step explanation:

new length=15+x

width=19+x

then area=(15+x)×(19+x)=285+15x+19x+x²=x²+34x+285 ft²

original area=15×19=285 ft²

then 285+98=x²+34x+285

or

x²+34x-98=0

x²+34x+17²=98+17²

(x+17)²=98+289=387

x+17=√387=3√43

x=3√43-17 ft

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in a football tournament at group stage there are five football teams in a group, Brazil, England, Scotland, Argentina and Franc
emmasim [6.3K]

Answer:

6/25

Step-by-step explanation:

Because england, scotland and france are european team and they make up 3/5 of all the teams, you multiply 3/5 by 2/5 because the south american teams are brazil and argentina which make up 2/5 of the total teams. So the probability that a european team will play a south american team is 3/5*2/5 which is 6/25

7 0
3 years ago
Twelve years ago, Jane was five times as old as Anne. In three years' time, Anne will be half Jane's
Soloha48 [4]

Answer:

Jane: 37 years old

Anne: 17

Step-by-step explanation:

Jane 12 years ago: 5x

Anne 12 years ago: x

In three years into the future:

Jane: 5x+15

Anne: x+15

CURRENT TIME:

Jane: 5x+12

Anne: x+12

Equation (using 3 years into the future ages): 5x+15=2(x+15)

5x+15=2x+30

(subtract 15 from both sides)

5x=2x+15

3x=15

x=5

Therefore, Jane is 37 years old and Anne is 17 years old.

3 0
3 years ago
Consider the transpose of Your matrix A, that is, the matrix whose first column is the first row of A, the second column is the
Zarrin [17]

Answer:The system could have no solution or n number of solution where n is the number of unknown in the n linear equations.

Step-by-step explanation:

To determine if solution exist or not, you test the equation for consistency.

A system is said to be consistent if the rank of a matrix (say B ) is equal to the rank of the matrix formed by adding the constant terms(in this case the zeros) as a third column to the matrix B.

Consider the following scenarios:

(1) For example:Given the matrix A=\left[\begin{array}{ccc}1&2\\3&4\end{array}\right], to transpose A, exchange rows with columns i.e take first column as first row and second column as second row as follows:

Let A transpose be B.

∵B=\left[\begin{array}{ccc}1&3\\2&4\end{array}\right]

the system Bx=0 can be represented in matrix form as:

\left[\begin{array}{ccc}1&3\\2&4\end{array}\right]\left[\begin{array}{ccc}x_{1} \\x_{2} \end{array}\right]=\left[\begin{array}{ccc}0\\0\end{array}\right] ................................eq(1)

Now, to determine the rank of B, we work the determinant of the maximum sub-square matrix of B. In this case, B is a 2 x 2 matrix, therefore, the maximum sub-square matrix of B is itself B. Hence,

|B|=(1*4)-(3*2)= 4-6 = -2 i.e, B is a non-singular matrix with rank of order (-2).

Again, adding the constant terms of equation 1(in this case zeros) as a third column to B, we have B_{0}:      

B_{0}=\left[\begin{array}{ccc}1&3&0\\4&2&0\end{array}\right]. The rank of B_{0} can be found by using the second column and third column pair as follows:

|B_{0}|=(3*0)-(0*2)=0 i.e, B_{0} is a singular matrix with rank of order 1.

Note: a matrix is singular if its determinant is = 0 and non-singular if it is \neq0.

Comparing the rank of both B and B_{0}, it is obvious that

Rank of B\neqRank of B_{0} since (-2)<1.

Therefore, we can conclude that equation(1) is <em>inconsistent and thus has no solution.     </em>

(2) If B=\left[\begin{array}{ccc}-4&5\\-8&10&\end{array}\right] is the transpose of matrix A=\left[\begin{array}{ccc}-4&-8\\5&10\end{array}\right], then

Then the equation Bx=0 is represented as:

\left[\begin{array}{ccc}-4&5\\-8&10&\end{array}\right]\left[\begin{array}{ccc}x_{1} \\x_{2} \end{array}\right]=\left[\begin{array}{ccc}0\\0\end{array}\right]..................................eq(2)

|B|= (-4*10)-(5*(-8))= -40+40 = 0  i.e B has a rank of order 1.

B_{0}=\left[\begin{array}{ccc}-4&5&0\\-8&10&0\end{array}\right],

|B_{0}|=(5*0)-(0*10)=0-0=0   i.e B_{0} has a rank of order 1.

we can therefor conclude that since

rank B=rank B_{0}=1,  equation(2) is <em>consistent</em> and has 2 solutions for the 2 unknown (X_{1} and X_{2}).

<u>Summary:</u>

  • Given an equation Bx=0, transform the set of linear equations into matrix form as shown in equations(1 and 2).
  • Determine the rank of both the coefficients matrix B and B_{0} which is formed by adding a column with the constant elements of the equation to the coefficient matrix.
  • If the rank of both matrix is same, then the equation is consistent and there exists n number of solutions(n is based on the number of unknown) but if they are not equal, then the equation is not consistent and there is no number of solution.
5 0
3 years ago
Javier rounded to the nearest half to estimate the product of 3 and two-fifths and Negative 3 and StartFraction 7 over 8 EndFrac
7nadin3 [17]

Answer:

The actual product is less than the estimate. The numbers differ by more than one.

Step-by-step explanation:

i took the test

8 0
3 years ago
Read 2 more answers
How did Spain and Portugal maintain control of their Latin American domains?
ioda
<span>through a rigid class structure: viceroys/colonial officers, Creoles, mestizos</span>
6 0
3 years ago
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