Question

Jacob, a chef who lives in the community, is really excited to launch his own cookbook. He found a local publishing house that charges him $25,500 per year in overhead (fixed costs), $14 per textbook in publishing costs, and royalties of 10% of the sales price. The company has indicated that they can print a maximum of 12,000 books per year. Jacob decided, with the publishing house, to sell the book for $25 each. a) Create a break-even graph that demonstrates fixed costs, total costs, total revenues, break-even point, and profit and loss areas. b) Determines the break-even point relative to volume and relative to revenue and calculates the break-even point as a percentage of maximum capacity. c) Calculate the new break-even point for volume and revenue if fixed costs increase by 15% per year, publishing costs increase by $6 per book and the selling price increases to $30.

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Answer to a math question Jacob, a chef who lives in the community, is really excited to launch his own cookbook. He found a local publishing house that charges him $25,500 per year in overhead (fixed costs), $14 per textbook in publishing costs, and royalties of 10% of the sales price. The company has indicated that they can print a maximum of 12,000 books per year. Jacob decided, with the publishing house, to sell the book for $25 each. a) Create a break-even graph that demonstrates fixed costs, total costs, total revenues, break-even point, and profit and loss areas. b) Determines the break-even point relative to volume and relative to revenue and calculates the break-even point as a percentage of maximum capacity. c) Calculate the new break-even point for volume and revenue if fixed costs increase by 15% per year, publishing costs increase by $6 per book and the selling price increases to $30.

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Esmeralda
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101 Answers
a) Pour créer le graphique de seuil de rentabilité, nous devons d'abord calculer les coûts totaux, les revenus totaux et le seuil de rentabilité.

Les coûts totaux comprennent les frais généraux annuels de 25 500 $ plus les coûts de publication de 14 $ par manuel multipliés par le nombre de livres vendus. Donc les coûts totaux sont :

Coûts totaux = Frais généraux annuels + (Coûts de publication par manuel x Nombre de livres vendus)

L'expression mathématique pour les coûts totaux est donc :

Coûts\ totaux = 25500 + (14\times Nombre\ de\ livres\ vendus)

Les revenus totaux sont calculés en multipliant le prix de vente par le nombre de livres vendus. Donc les revenus totaux sont :

Revenus totaux = Prix de vente x Nombre de livres vendus

L'expression mathématique pour les revenus totaux est donc :

Revenus\ totaux = 25 \times Nombre\ de\ livres\ vendus

Le seuil de rentabilité est le point où les revenus totaux sont égaux aux coûts totaux. Donc, on a :

Revenus totaux = Coûts totaux

Maintenant, nous pouvons tracer le graphique en utilisant ces informations.

b) Pour déterminer le seuil de rentabilité relativement au volume, nous devons résoudre l'équation :

Coûts\ totaux = Prix\ de\ vente \times Nombre\ de\ livres\ vendus

En remplaçant les variables par leurs valeurs, nous avons :

25500 + 14 \times Nombre\ de\ livres\ vendus = 25 \times Nombre\ de\ livres\ vendus

En simplifiant cette équation, nous obtenons :

25500 = 11 \times Nombre\ de\ livres\ vendus

Pour résoudre cette équation pour Nombre de livres vendus, nous divisons les deux côtés par 11 :

Nombre\ de\ livres\ vendus = \frac{25500}{11} = 2318.18

Donc, le seuil de rentabilité relativement au volume est d'environ 2319 livres.

Pour déterminer le seuil de rentabilité relativement aux revenus, nous devons utiliser la même équation et isoler Prix de vente :

Prix\ de\ vente = \frac{Coûts\ totaux}{Nombre\ de\ livres\ vendus}

En remplaçant les variables par leurs valeurs, nous avons :

Prix\ de\ vente = \frac{25500 + 14 \times Nombre\ de\ livres\ vendus}{Nombre\ de\ livres\ vendus}

Maintenant, nous pouvons substituer la valeur de Nombre de livres vendus que nous avons obtenue précédemment :

Prix\ de\ vente = \frac{25500 + 14 \times 2318.18}{2318.18} = 25

Donc, le seuil de rentabilité relativement aux revenus est de 25 $.

Pour calculer le seuil de rentabilité en pourcentage de la capacité maximale, nous divisons le seuil de rentabilité relativement au volume par la capacité maximale et multiplions par 100 :

Seuil\ de\ rentabilité\ en\ pourcentage = \frac{2319}{12000} \times 100 \approx 19.33\%

c) Si les coûts fixes augmentent de 15% par année, alors les nouveaux frais généraux annuels seront de :

Nouveaux frais généraux annuels = Frais généraux annuels + (Frais généraux annuels x Taux d'augmentation)

En utilisant les valeurs données, nous avons :

Nouveaux frais généraux annuels = 25500 + (25500 x 0.15) = 29325

Si les coûts de publication augmentent de 6 $ par livre, alors les nouveaux coûts de publication par livre seront de :

Nouveaux coûts de publication par livre = Coûts de publication par livre + Augmentation des coûts de publication par livre

En utilisant les valeurs données, nous avons :

Nouveaux coûts de publication par livre = 14 + 6 = 20

Si le prix de vente augmente à 30 $, alors le nouveau prix de vente sera de :

Nouveau prix de vente = Prix de vente + Augmentation du prix de vente

En utilisant les valeurs données, nous avons :

Nouveau prix de vente = 25 + 5 = 30

Maintenant, nous pouvons répéter les étapes précédentes pour calculer le nouveau seuil de rentabilité relativement au volume et relativement aux revenus avec ces nouvelles valeurs.

Nouveau\ seuil\ de\ rentabilité\ relativement\ au\ volume = \frac{Nouveaux\ frais\ généraux\ annuels}{Prix\ de\ vente\ -\ Nouveaux\ coûts\ de\ publication\ par\ livre} = \frac{29325}{30 - 20} = 2932.5

Nouveau\ seuil\ de\ rentabilité\ relativement\ aux\ revenus = \frac{Nouveaux\ frais\ généraux\ annuels}{Nouveau\ prix\ de\ vente} = \frac{29325}{30} = 977.5

Le nouveau seuil de rentabilité relativement au volume est d'environ 2933 livres, et le nouveau seuil de rentabilité relativement aux revenus est d'environ 977.5 $.

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