Find two positive real numbers with the given sum whose product is a maximum. The sum of the first and twice the second is 24

1 answer:

5 0

Y+2x=24
product is max
xy=max

we have
y+2x=24
minus 2x both sides
y=24-2x
sub that for y in xy=max
(24-2x)(x)=max
-2x^2+24x=max
we have a parabola opening down
the coordinates of the vertex is our answer

the x value of the vertex is -b/2a where
ax^2+bx+c=y

-24/2(-2)=-24/-4=6
xvalue=6

we can just sub
y=24-2x
y=24-2(6)
y=24-12
y=12
the numbers are 6 and 12
the product is 72

the numbers are 6 and 12

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Geometry question, need help!

Anna35 [415]

Answer:

x = 7.5

Step-by-step explanation:

We first labelled the diagram we get

DC = 8

AD = 10

CE = 6

BE = x

∴ AC =AD + DC = 18

∴ BC = BE + ED = x + 6

To Find :

BE = x =?

Solution:

Let DE || AC

In Δ ABC and Δ DEC

∠A ≅ ∠D …………..{corresponding angles ∵ DE || AB }

∠B ≅ ∠E ..............{corresponding angles ∵ DE || AB }

∠C ≅ ∠C ……….....{Reflexive Property}

Δ ABC ~ Δ DEC ….{Angle-Angle-Angle Similarity test}

If two triangles are similar then their sites are in proportion.

$\frac{AB}{DE} =\frac{BC}{EC} =\frac{AC}{CD}\ \textrm{corresponding sides of similar triangles are in proportion}\\$

On substituting the given values we get

∴ $\frac{8}{18} = \frac{6}{6+x} \\\\8(6+x) = 6\times 18\\\\\textrm{using distributive property we get}\\48+8x=108\\8x=108-48\\x=\frac{60}{8}\\ \\x=7.5\\\\\therefore x =7.5$