Resources

Resources

Compare linear and exponential growth

c=\frac{4 b}{\sqrt[3]{d}}

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The formula gives the capsize screening value,

c

, for a sailboat with a beam

b

feet long and that displaces

d

pounds of water. Higher capsize screening values suggest that a sailboat is more stable. Which of the following equations correctly gives the displacement in terms of the capsize screening value and the beam length?

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1

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d=\frac{(4 b)^{3}}{c}

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d=\frac{c^{3}}{4 b}

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d=\left(\frac{4 b}{c}\right)^{3}

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d=\left(\frac{c}{4 b}\right)^{3}

F=\frac{9}{5}(K-273)+32

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In order to convert a temperature in Kelvin,

K

, into a temperature in degrees Fahrenheit,

F

, the given equation can be used. Which of the following equations correctly expresses the temperature in Kelvin in terms of the temperature in degrees Fahrenheit?

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1

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K=\frac{9}{5}(F-273)+32

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K=\left(\frac{5}{9} F-32\right)+27

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K=\frac{5}{9}(F-32)+273

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K=\frac{5(F-32)+273}{9}

r=\sqrt{x^{2}+y^{2}}

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The equation gives the quantity

r

in terms of the quantities

x

y

. Which of the following equations gives a possible value of

x

r

y

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1

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x=\sqrt{r^{2}}-y^{2}

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x=\sqrt{r^{2}-y^{2}}

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x=\sqrt{r^{2}+y^{2}}

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x=r-y

Q=\sqrt{\frac{2 K D}{F}}

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For companies that monitor the inventory of a product, the equation gives

Q

, the quantity to order of the product as a function of

K

, the ordering cost,

D

, the annual demand for the product and

F

, the average holding cost of the product. Which of the following equations correctly gives the annual demand for the product in terms of the quantity to order, the ordering cost, the annual demand, and the average holding cost for the product?

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1

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D=\sqrt{\frac{2 F Q}{K}}

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D=\sqrt{\frac{F Q}{2 K}}

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D=\frac{2 F Q^{2}}{K}

p^{2}+2 p q+q^{2}=1

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In population genetics, the HardyWeinberg law, shown, describes the relative frequency of two alleles,

p

q

. The values of

p

q

are always between

0

1

. Which of the following correctly expresses

p

q

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1

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(p+q)^{2}=1

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(p+q)^{2}=0

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p=-q+1

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p=-q

3 y^{2}+5 y-10=0

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What are the solutions to the given equation?

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1

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y=-\frac{5}{2}-\frac{\sqrt{145}}{2}

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y=-\frac{5}{2}+\frac{\sqrt{145}}{2}

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y=-\frac{5}{6}-\frac{\sqrt{145}}{6}

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y=-\frac{5}{6}+\frac{\sqrt{145}}{6}

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y=\frac{5}{6}-\frac{\sqrt{145}}{6}

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y=\frac{5}{6}+\frac{\sqrt{145}}{6}

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y=\frac{5}{2}-\frac{\sqrt{145}}{2}

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y=\frac{5}{2}+\frac{\sqrt{145}}{2}

k^{2}=m^{2}+n^{2}

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For any right triangle, the given equation relates the length of the hypotenuse,

k

, to the lengths of the other two sides of the triangle,

m

n

. Which of the following equations correctly gives

m

k

n

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1

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m=k-n

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m=\sqrt{k^{2}}-n^{2}

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m=\sqrt{k^{2}-n^{2}}

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m=\sqrt{k^{2}+n^{2}}

V=\frac{4}{3} \pi r^{3}

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V

, of a sphere of radius,

r

, can be found with the given equation. Which of the following correctly shows the sphere's radius in terms of its volume?

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1

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r=\frac{3}{4} \pi V^{\frac{1}{3}}

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r=\frac{4}{3} \pi V^{\frac{1}{3}}

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r=\left(\frac{3 V}{4 \pi}\right)^{\frac{1}{3}}

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r=\left(\frac{4 V}{3 \pi}\right)^{\frac{1}{3}}

\frac{2}{3} x^{2}-\frac{1}{2} x-\frac{3}{4}=0

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What is the sum of the solutions to the given equation?

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1

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-\frac{3}{2}

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-\frac{3}{4}

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0

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\frac{3}{4}

(1-a)+3(1-a)^{2}=0

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What are the solutions to the given equation?

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1

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a=1

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a=\frac{4}{3}

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a=1

a=\frac{4}{3}

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a=0

a=-\frac{1}{3}

s

s(x)=(x-4)(x-5)^{2}

h

h(x)=(x-a) \cdot s(x)

. For some constant

a,(x-a)^{3}

h

s(a)

\begin{array}{l} p(n)=\left(n^{3}-12 n^{2}+47 n-60\right) \\ q(n)=(n+13)(n-4) \end{array}

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p

q

are defined. One of the functions has a zero at

n=5

. What is the value of

p(5)+q(5) ?

An investment lost approximately

5 \%

of the balance each month for the past year. The amount of the investment on January

1^{\text {st }}

of last year was

\$ 10,000

. Which of the following functions,

I

, models the amount of the investment (in thousands of dollars) at the end of month,

n

1 \leq n \leq 12

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1

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I(n)=10 \cdot(1.05)^{n}

thousand dollars

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I(n)=10-0.05 n

thousand dollars

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I(n)=10 \cdot(0.95)^{n}

thousand dollars

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I(n)=10+0.95 n

thousand dollars

The population of City Z was

50

000

1983

1983

2013

, the population of City Z doubled every

5

years. Which of the following best models

P

, the population of City Z from

1983

2013, t

1983

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1

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P=50,000(2)^{\frac{t}{5}}

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P=50,000(2)^{\mathrm{Br}}

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P=10,000 t+50,000

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P=20,000 t+50,000

z=20 i+26

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What are the real and imaginary parts of

z

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1

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\operatorname{Re}(z)=20

\operatorname{Im}(z)=26

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\operatorname{Re}(z)=26

\operatorname{Im}(z)=20 i

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\operatorname{Re}(z)=20 i

\operatorname{Im}(z)=26

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\operatorname{Re}(z)=26

\operatorname{Im}(z)=20

A monkey is swinging from a tree. On the first swing, she passes through an arc whose length is

24 \mathrm{~m}

. With each swing, she travels along an arc that is half as long as the arc of the previous swing.

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Which expression gives the total length the monkey swings in her first

n

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1

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24 \cdot \frac{\left(1-1.5^{n}\right)}{-0.5}

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24 \cdot \frac{\left(1-1.5^{n}\right)}{0.5}

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24 \cdot \frac{\left(1-0.5^{n}\right)}{-0.5}

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24 \cdot \frac{\left(1-0.5^{n}\right)}{0.5}

z=3-6 i

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What are the real and imaginary parts of

z

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1

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\begin{array}{l} \operatorname{Re}(z)=3 \text { and } \\ \operatorname{Im}(z)=-6 \end{array}

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\begin{array}{l} \operatorname{Re}(z)=-6 \text { and } \\ \operatorname{Im}(z)=3 \end{array}

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\begin{array}{l} \operatorname{Re}(z)=3 \text { and } \\ \operatorname{Im}(z)=-6 i \end{array}

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\begin{array}{l} \operatorname{Re}(z)=-6 i \text { and } \\ \operatorname{Im}(z)=3 \end{array}

z=32+41.9 i

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What are the real and imaginary parts of

z

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1

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\begin{array}{l} \operatorname{Re}(z)=32 \text { and } \\ \operatorname{Im}(z)=41.9 i \end{array}

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\begin{array}{l} \operatorname{Re}(z)=32 \text { and } \\ \operatorname{Im}(z)=41.9 \end{array}

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\begin{array}{l} \operatorname{Re}(z)=41.9 \text { and } \\ \operatorname{Im}(z)=32 \end{array}

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\begin{array}{l} \operatorname{Re}(z)=41.9 i \text { and } \\ \operatorname{Im}(z)=32 \end{array}

7 x^{6}

\left(3 x^{2}\right)\left(4 x^{4}\right)

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Olivia factored

7 x^{6}

\left(7 x^{2}\right)\left(x^{3}\right)

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Which of them factored

7 x^{6}

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1

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(B) Only Olivia

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(C) Both Will and Olivia

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(D) Neither Will nor Olivia

Fernando factored

45 y^{6}

\left(9 y^{3}\right)\left(5 y^{3}\right)

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45 y^{6}

(3 y)\left(15 y^{5}\right)

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Which of them factored

45 y^{6}

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1

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(A) Only Fernando

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(C) Both Fernando and Salma

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(D) Neither Fernando nor Salma

Which of the equations are true identities?

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(x-2 y)^{2}=x^{2}-4 x y+4 y^{2}

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(m+1)^{2}-m^{2}=1

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1

\newline

\newline

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\mathrm{A}

\mathrm{B}

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\mathrm{A}

\mathrm{B}

We want to factor the following expression:

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9 x^{4}+4 y^{2}

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Which pattern can we use to factor the expression?

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U

V

are either constant integers or single-variable expressions.

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1

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(U+V)^{2}

(U-V)^{2}

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(U+V)(U-V)

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(C) We can't use any of the patterns.

We want to factor the following expression:

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16 x^{4}-25

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Which pattern can we use to factor the expression?

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U

V

are either constant integers or single-variable expressions.

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1

\newline

(U+V)^{2}

(U-V)^{2}

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(U+V)(U-V)

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(C) We can't use any of the patterns.

We want to factor the following expression:

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x^{4}+3 x^{2} y+9 y^{2}

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Which pattern can we use to factor the expression?

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U

V

are either constant integers or single-variable expressions.

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1

\newline

(U+V)^{2}

(U-V)^{2}

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(U+V)(U-V)

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(C) We can't use any of the patterns.

We want to factor the following expression:

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(x+5)^{4}+2(x+5)^{2}-1

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Which pattern can we use to factor the expression?

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U

V

are either constant integers or single-variable expressions.

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1

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(U+V)^{2}

(U-V)^{2}

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(U+V)(U-V)

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(C) We can't use any of the patterns.

We want to factor the following expression:

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(x-3)^{2}-10(x-3)+25

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Which pattern can we use to factor the expression?

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U

V

are either constant integers or single-variable expressions.

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1

\newline

(U+V)^{2}

(U-V)^{2}

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(U+V)(U-V)

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(C) We can't use any of the patterns.

We want to factor the following expression:

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(x+7)^{2}+2 y(x+7)+y^{2}

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Which pattern can we use to factor the expression?

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U

V

are either constant integers or single-variable expressions.

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1

\newline

(U+V)^{2}

(U-V)^{2}

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(U+V)(U-V)

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(C) We can't use any of the patterns.

We want to factor the following expression:

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9 x^{6}+6 x^{3} y+y^{2}

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Which pattern can we use to factor the expression?

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U

V

are either constant integers or single-variable expressions.

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1

\newline

(U+V)^{2}

(U-V)^{2}

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(U+V)(U-V)

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(C) We can't use any of the patterns.

We want to factor the following expression:

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x^{4}+10 x+25

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Which pattern can we use to factor the expression?

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U

V

are either constant integers or single-variable expressions.

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1

\newline

(U+V)^{2}

(U-V)^{2}

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(U+V)(U-V)

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(C) We can't use any of the patterns.

We want to factor the following expression:

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4 x^{2}-9 y^{6}

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Which pattern can we use to factor the expression?

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U

V

are either constant integers or single-variable expressions.

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1

\newline

(U+V)^{2}

(U-V)^{2}

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(U+V)(U-V)

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(C) We can't use any of the patterns.

We want to factor the following expression:

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3 x^{2}-16 y^{5}

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Which pattern can we use to factor the expression?

\newline

U

V

are either constant integers or single-variable expressions.

\newline

1

\newline

(U+V)^{2}

(U-V)^{2}

\newline

(U+V)(U-V)

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(C) We can't use any of the patterns.

We want to factor the following expression:

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(x+1)^{2}-4 y^{2}

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Which pattern can we use to factor the expression?

\newline

U

V

are either constant integers or single-variable expressions.

\newline

1

\newline

(U+V)^{2}

(U-V)^{2}

\newline

(U+V)(U-V)

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(C) We can't use any of the patterns.

We want to factor the following expression:

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(x-2)^{4}-(y+3)^{4}

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Which pattern can we use to factor the expression?

\newline

U

V

are either constant integers or single-variable expressions.

\newline

1

\newline

(U+V)^{2}

(U-V)^{2}

\newline

(U+V)(U-V)

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(C) We can't use any of the patterns.

We want to factor the following expression:

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(x-1)^{2}+(x-1)+4

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Which pattern can we use to factor the expression?

\newline

U

V

are either constant integers or single-variable expressions.

\newline

1

\newline

(U+V)^{2}

(U-V)^{2}

\newline

(U+V)(U-V)

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(C) We can't use any of the patterns.

(b d)^{\frac{1}{4}} \cdot\left(\frac{b}{d}\right)^{4}

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Which of the following expressions is equivalent to the given expression assuming

d

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1

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b^{\frac{17}{4}} d^{\frac{17}{4}}

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b^{\frac{17}{4}} d^{-\frac{15}{4}}

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b^{2}

\newline

\frac{b}{d}

\left(\frac{1}{9}\right)^{-\frac{3}{2}} \cdot\left(\frac{1}{8}\right)^{\frac{2}{3}}

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What is the value of the given expression?

6^{\frac{1}{6}} \cdot 2^{\frac{5}{6}} \cdot 3^{\frac{7}{6}}

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Which of the following is equivalent to the given expression?

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1

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6 \sqrt[3]{3}

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2^{\frac{5}{36}} \cdot 3^{\frac{7}{36}}

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36^{\frac{35}{216}}

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\sqrt[3]{6^{13}}

\sqrt{16 x^{2} y^{16}+25 x^{16} y^{2}}

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x

y

are positive, which of the following is equivalent to the given expression?

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1

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4 x y^{8}+5 x^{8} y

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4 x y^{4}+5 x^{4} y

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x y \sqrt{16 y^{14}+25 x^{14}}

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x^{2} y^{2} \sqrt{16 y^{14}+25 x^{14}}

\left(x^{\frac{7}{4}} \cdot x^{4}\right)^{4}

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Which of the following expressions is equivalent to the given expression assuming

x

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1

\newline

x \frac{39}{4}

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x^{11}

\newline

x^{23}

\newline

x^{28}

W(x)=0.25 x^{2}+0.2 x

V(x)=100 x+40

, which of the following is equal to

W(x) \cdot V(x)

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1

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0.25 x^{2}+99.8 x+40

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25 x^{3}+30 x^{2}+8 x

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0.25 x^{2}+20 x+40

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25 x^{3}+10 x^{2}+20 x+8

(-5 z)-(2-z) \cdot(3-z)

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Which of the following expressions is equivalent to the given expression?

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1

\newline

-z^{2}-10 z-6

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-z^{2}-4 z+6

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-z^{2}-6

\newline

z^{2}-10 z-6

\left(2+\frac{1}{3} d\right)\left(5 d+\frac{1}{3} d^{2}\right)

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Which of the following expressions is equivalent to the given expression?

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1

\newline

2+2 d^{2}

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2+\frac{16}{3} d+\frac{1}{3} d^{2}

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2+\frac{5}{3} d^{2}+\frac{1}{9} d^{3}

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10 d+\frac{7}{3} d^{2}+\frac{1}{9} d^{3}

Which of the following is the sum of

p-2

(p+3)(p-6)

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1

\newline

3 p-5

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2 p^{2}-20

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p^{2}-2 p-20

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p^{3}-5 p^{2}-12 p+36

f(x)=x^{2}-2

g(x)=x-3

, which of the following is equivalent to

f(x)-[g(x)]^{2} ?

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1

\newline

-7

\newline

-11

\newline

6 x-11

\newline

x^{4}-2 x^{3}+3 x^{2}-2 x+1

\begin{array}{l} f(x)=3 x^{2}+4 x-2 \\ g(x)=7 x^{2}-b x+3 \end{array}

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f(x)-g(x)=-4 x^{2}+6 x-5

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for all values of

x

b

is a constant, then what is the value of

b

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1

\newline

-10

\newline

-2

\newline

2

\newline

10

4 x^{2}+28 x+49

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Which of the following is equivalent to the given expression?

\newline

1

\newline

(2 x+7)^{2}

\newline

(2 x+49)^{2}

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(4 x+7)^{2}

\newline

(4 x+49)^{2}

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\left(y^{2}-t^{2}\right)(y+k)

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can be written as

\newline

y^{3}+36 y^{2}-9 y+s

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t, k

s

are constants. What is the value of

s

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1

\newline

-324

\newline

-54

\newline

54

\newline

324

2 x(x-2)+3\left(4-x^{2}\right)

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Which of the following is equivalent to the given expression?

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1

\newline

(6+x)(2-x)

\newline

(x-6)(x+2)

\newline

x^{2}+10

\newline

x^{2}-4 x+12

m^{2}+p^{2}=x

4 m p=y

, which of the following is equivalent to

4 x+2 y

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1

\newline

(2 m+p)^{2}

\newline

(2 m+2 p)^{2}

\newline

(4 m+2 p)^{2}

\newline

(4 m+8 p)^{2}

(w-3)^{5}-(w-3)^{2}

\newline

Which of the following is equivalent to the given expression?

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1

\newline

(w-3)^{3}

\newline

w^{5}-w^{2}-252

\newline

(w-3)^{2}(w-3)^{3}

\newline

(w-3)^{2}\left((w-3)^{3}-1\right)

...

...