The ratio of koi to goldfish in a pond is 4:9. If there are a total of 39 fish in the pond, how many of these fish are koi?

Mathematics

2 answers:

Veseljchak [2.6K]3 years ago

6 0

12 koi to x goldfish

pickupchik [31]3 years ago

5 0

12 kio to × goldfish

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Factoring with Diagrams

Finger [1]

Answer:

(x - 7)(x - 4)(x + 4).

Step-by-step explanation:

A(x) = x3 – 7x² – 16x +112

Dividing by x-7:

x^2 - 16 <-------- Quotient

-----------------------------

x-7)x^3 - 7x^2 - 16x + 112

x^3 - 7x^2

0 - 16x + 112

- 16x + 112

..............

so A(x) = (x - 7)(x^2 - 16) x^2 - 16 is the difference of 2 squares so we have:

(x - 7)(x - 4)(x + 4).

Checking by expanding the brackets:

(x - 7)(x - 4)(x + 4)

= x(x - 4)(x + 4) - 7(x - 4)(x + 4)

= x(x^2 - 16) - 7(x^2 - 16)

= x^3 - 16x - 7x^2 + 114

= x^3 - 7x^2 - 16x + 112

7 0

3 years ago

At a point on the ground 80 ft from the base of a tree, the distance to the top of the tree is 11 ft more than 2 times the heigh

Sergio039 [100]

Let h be the height of the tree and d the distance to the top of the tree from the point on the ground. Draw a diagram to visualize the situation:

Since the distance to the top of the tree is 11 ft more than two times the height, then:

$d=2h+11$

Use the Pythagorean Theorem to relate the length of the sides of the right triangle:

$\begin{gathered} h^2+80^2=d^2 \\ \Rightarrow h^2+6400=(2h+11)^2 \\ \operatorname{\Rightarrow}h^2+6400=(2h)^2+2(11)(2h)+11^2 \\ \operatorname{\Rightarrow}h^2+6400=4h^2+44h+121 \end{gathered}$

Notice that we have obtained a quadratic equation in terms of h. Write it in standard form and use the quadratic formula to solve for h:

$\begin{gathered} \Rightarrow0=4h^2+44h+121-h^2-6400 \\ \Rightarrow0=4h^2-h^2+44h+121-6400 \\ \Rightarrow0=3h^2+44h-6279 \\ \Rightarrow3h^2+44h-6279=0 \\ \\ \Rightarrow h=\frac{-44\pm\sqrt{(44)^2-4(3)(-6279)}}{2(3)} \\ \\ \therefore h_1=39 \\ h_2=-53.666.. \end{gathered}$

Since the height of the tree must be positive, the only solution is h=39ft. To the nearest foot, the height of the tree is 39.

Therefore, the height of the tree is 39 ft.