1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
m_a_m_a [10]
3 years ago
11

Find the area of a triangle bounded by the y-axis, the line f(x)=9−4/7x, and the line perpendicular to f(x) that passes through

the origin.
Area =
Mathematics
1 answer:
Setler79 [48]3 years ago
8 0

<u>ANSWER:  </u>

The area of the triangle bounded by the y-axis is  \frac{7938}{4225} \sqrt{65} \text { unit }^{2}

<u>SOLUTION:</u>

Given, f(x)=9-\frac{-4}{7} x

Consider f(x) = y. Hence we get

f(x)=9-\frac{-4}{7} x --- eqn 1

y=9-\frac{4}{7} x

On rewriting the terms we get

4x + 7y – 63 = 0

As the triangle is bounded by two perpendicular lines, it is an right angle triangle with y-axis as hypotenuse.

Area of right angle triangle = \frac{1}{ab} where a, b are lengths of sides other than hypotenuse.

So, we need find length of f(x) and its perpendicular line.

First let us find perpendicular line equation.

Slope of f(x) = \frac{-x \text { coefficient }}{y \text { coefficient }}=\frac{-4}{7}

So, slope of perpendicular line = \frac{-1}{\text {slope of } f(x)}=\frac{7}{4}

Perpendicular line is passing through origin(0,0).So by using point slope formula,

y-y_{1}=m\left(x-x_{1}\right)

Where m is the slope and \left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)

y-0=\frac{7}{4}(x-0)

y=\frac{7}{4} x --- eqn 2

4y = 7x

7x – 4y = 0  

now, let us find the vertices of triangle, one of them is origin, second one is point of intersection of y-axis and f(x)

for points on y-axis x will be zero, to get y value, put x =0 int f(x)

0 + 7y – 63 = 0

7y = 63

y = 9

Hence, the point of intersection is (0, 9)

Third vertex is point of intersection of f(x) and its perpendicular line.

So, solve (1) and (2)

\begin{array}{l}{9-\frac{4}{7} x=\frac{7}{4} x} \\\\ {9 \times 4-\frac{4 \times 4}{7} x=7 x} \\\\ {36 \times 7-16 x=7 \times 7 x} \\\\ {252-16 x=49 x} \\\\ {49 x+16 x=252} \\\\ {65 x=252} \\\\ {x=\frac{252}{65}}\end{array}

Put x value in (2)

\begin{array}{l}{y=\frac{7}{4} \times \frac{252}{65}} \\\\ {y=\frac{441}{65}}\end{array}

So, the point of intersection is \left(\frac{252}{65}, \frac{441}{65}\right)

Length of f(x) is distance between \left(\frac{252}{65}, \frac{441}{65}\right) and (0,9)

\begin{aligned} \text { Length } &=\sqrt{\left(0-\frac{252}{65}\right)^{2}+\left(9-\frac{441}{65}\right)^{2}} \\ &=\sqrt{\left(\frac{252}{65}\right)^{2}+0} \\ &=\frac{252}{65} \end{aligned}

Now, length of perpendicular of f(x) is distance between \left(\frac{252}{65}, \frac{441}{65}\right) \text { and }(0,0)

\begin{aligned} \text { Length } &=\sqrt{\left(0-\frac{252}{65}\right)^{2}+\left(0-\frac{441}{65}\right)^{2}} \\ &=\sqrt{\left(\frac{252}{65}\right)^{2}+\left(\frac{441}{65}\right)^{2}} \\ &=\frac{\sqrt{(12 \times 21)^{2}+(21 \times 21)^{2}}}{65} \\ &=\frac{63}{65} \sqrt{65} \end{aligned}

Now, area of right angle triangle = \frac{1}{2} \times \frac{252}{65} \times \frac{63}{65} \sqrt{65}

=\frac{7938}{4225} \sqrt{65} \text { unit }^{2}

Hence, the area of the triangle is \frac{7938}{4225} \sqrt{65} \text { unit }^{2}

You might be interested in
You make 42,500 and your spouse makes 37750 . using the 20% to 25% range , calculate the least and the greatest amount you shoul
Zepler [3.9K]
If you make 42,500 and your spouse makes 37750, then using the 20% to 25% range, the least and the greatest amount you should spend on housing is $50,00. The correct answer to the following given statement above is 50 dollars.
8 0
3 years ago
Find the midpoint of segment EF if E (5,-2) F(1,-8) URGENT
Elina [12.6K]

Answer

(3, -5)

Step-by-step explanation:

hope it helps :)

4 0
3 years ago
Please take your time but I really need anyone's help!!!!
stiv31 [10]

Answer:

C) 5 3√2

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
Consider the following three points.(-5,2), (0,6), (6,4)Step 3 of 3: Determine whether the three points are collinear or not col
Anni [7]

Step 1

Collinear points are a set of three or more points that exist on the same straight line. Collinear points may exist on different planes but not on different lines.

Step 2

Graph the points; (-5,2),(0,6),(6,4)

Step 3

Conclude based on step 2

Since the points are not a straight line, we can conclude that the 3 points are not collinear.

3 0
1 year ago
At which root does the graph of f(x) = (x + 4)(x + 7,5) cross the x-axis?<br> -7<br> -4<br> 4<br> 7
Alona [7]

Answer:

Answer: -7

Step-by-step explanation:

5 0
2 years ago
Other questions:
  • Which of the following shows the graph of y = 4* + 3?
    7·1 answer
  • What is the quotient of 2/9 divided by 4/15
    13·2 answers
  • I WILL GIVE BRAINLIEST ANSWER TO THE CORRECT ANSWER!! PLEASE ASAP
    6·1 answer
  • Expand 4(-2y + 3). 2 y + 7 -8 y + 12 2 y + 12 -8 y + 3
    15·1 answer
  • | x | = 6 <br><br> What are the two values that x can equal?
    15·2 answers
  • Evaluate the function requested. Write your answer as a fraction in lowest terms. find cos B
    7·1 answer
  • Given the equation of y=1/4x, write an equation of a line that is steeper than the given line
    10·1 answer
  • 5. Find x and UV if V is between<br> U and w, with UV = 3x + 7,<br> VW = 4x + 5 and UW = x + 96
    5·1 answer
  • 9x9?????????????????????????
    10·2 answers
  • Concept of place value, ordering and rounding (in small and easy words) ​
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!