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Pepsi [2]
3 years ago
11

The line $y = b-x$ with $0 < b < 4$ intersects the $y$-axis at $P$ and the line $x=4$ at $S$. If the ratio of the area of

triangle $QRS$ to the area of triangle $QOP$ is 9:25, what is the value of $b$? Express the answer as a decimal to the nearest tenth.

Mathematics
2 answers:
olga55 [171]3 years ago
6 0

Answer:

b=2.5

Step-by-step explanation:

The given line has equation: y=b-x

The x-intercept is (b,0)

The y-intercept is (0,b)

The area of triangle QOP =\frac{1}{2} |OQ|*|OP|=

=\frac{1}{2}b*b

=\frac{1}{2}b^2

The area of triangle QRS =\frac{1}{2} |RQ|*|RS|=

=\frac{1}{2}(4-b)*(b-4)

The area of triangle QRS and QOP are in the ratio 9:25

We must negate the area under the x-axis.

\implies \frac{- \frac{1}{2}(4-b)(b-4)}{\frac{1}{2}b^2} =\frac{9}{25}

\implies \frac{-(4-b)(b-4)}{b^2} =\frac{9}{25}

\implies {-(4-b)(b-4) =\frac{9}{25}b^2

\implies b^2-8b+16=\frac{9}{25}b^2

\implies 25b^2-200b+400=9b^2

\implies 25b^2-9b^2-200b+400=0

\implies 16b^2-200b+400=0

\implies 2b^2-25b+50=0

(2b-5)(b-10)=0

b=2.5\:or\:b=10

But 0<b<4

Therefore b=2.5

VERIFY

The area of triangle QOP

=\frac{1}{2}*2.5*2.5=3.125

The area of triangle QRS

=|\frac{1}{2}(4-2.5)*(2.5-4)|=|-1.125|=1.125

We must take absolute value because this area is below the x-axis.

\frac{1.125}{3.125} =\frac{9}{25}

Softa [21]3 years ago
3 0

Answer:

b=2.5

Step-by-step explanation:

The line y=b-x intersects the x-axis at the point where 0 = b-x, or x=b. So, we seek the x-coordinate of point Q.

Since the y-axis is parallel to the line x = 4, we see that angle QSR = angle QPO. Also QOP = QRS = 90 degrees. Thus triangle QOP is similiar to triangle QRS, so [QRS]/[QOP] =(QR/QO)^2, which means we have (QR/QO)^2/9/25, so QR/QO= 3/5. Since QR + QO= 4, we have 3/5(QO + QO) = 4, and QO =4 * 5/8= 5/2. Therefore, the x-coordinate of Q is 5/2 = 2.5

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